Answer:
If [tex]n^2[/tex] is divisible by 3, the n is also divisible by 3.
Step-by-step explanation:
We will prove this with the help of contrapositive that is we prove that if n is not divisible by 3, then, [tex]n^2[/tex] is not divisible by 3.
Let n not be divisible by 3. Then [tex]\frac{n}{3}[/tex] can be written in the form of fraction [tex]\frac{x}{y}[/tex], where x and y are co-prime to each other or in other words the fraction is in lowest form.
Now, squaring
[tex]\frac{n^2}{9} = \frac{x^2}{y^2}[/tex]
Thus,
[tex]n^2 = \frac{9x^2}{y^2}[/tex]
[tex]\frac{n^2}{3} = \frac{3x^2}{y^2}[/tex]
It can be clearly seen that the fraction [tex]\frac{3x^2}{y^2}[/tex] is in lowest form.
Hence, [tex]n^2[/tex] is not divisible by 3.
Thus, by contrapositivity if [tex]n^2[/tex] is divisible by 3, the n is also divisible by 3.
Use the Euclidean algorithm to determine the greatest common divisor of 2288 and 4875
Answer:
13
Step-by-step explanation:
We have to find greatest common divisor of two number 2288 and 4875.
Now, greatest common divisor of two number is defined as the highest common factor that divides both the number.
We can use the Euclidean algorithm to do so.
Since 4875 is the larger of the two number
4875 ÷ 2288: Quotient = 2, Remainder = 299
2288 ÷ 299: Quotient = 7, Remainder = 195
299 ÷ 195: Quotient = 1, Remainder = 104
195 ÷ 104: Quotient = 1, Remainder = 91
104 ÷ 91: Quotient = 1, Remainder = 13
91 ÷ 13: Quotient =7, Remainder = 0
Hence, we stop here and the greatest common divisor = 13
Final answer:
The greatest common divisor of 2288 and 4875 is 1.
Explanation:
The Euclidean algorithm is used to determine the greatest common divisor (GCD) of two numbers. To find the GCD of 2288 and 4875, we will use the Euclidean algorithm as follows:
Divide 4875 by 2288, giving a quotient of 2 and a remainder of 299.
Divide 2288 by 299, giving a quotient of 7 and a remainder of 35.
Divide 299 by 35, giving a quotient of 8 and a remainder of 19.
Divide 35 by 19, giving a quotient of 1 and a remainder of 16.
Divide 19 by 16, giving a quotient of 1 and a remainder of 3.
Divide 16 by 3, giving a quotient of 5 and a remainder of 1.
The Euclidean algorithm ends when we reach a remainder of 1. The GCD of 2288 and 4875 is the previous remainder, which is 1.
So, the greatest common divisor of 2288 and 4875 is 1.
assume that the number of production defects is 28 and that 14 of these are classified as major defects, 8 are classified as design defects, and 11 were neither major defects nor design defects. How many of the design defects were major?
Answer: 5
Step-by-step explanation:
Let A denotes the number of major defects and B denotes the number of design defect.
By considering the given information, we have
[tex]U=28\ ;\ n(A)=14\ ;\ n(B)=8\ \ ;\ n(A^c\cap B^c)=1[/tex]
Now, the number of major defects or design defects:
[tex]n(A\cup B)=U-n(A^c\cap B^c)=28-11=17[/tex]
Also,
[tex]n(A\cap B)=n(A)+n(B)-n(A\cup B)\\\\\Rightarrow\ n(A\cap B)=14+8-17=5[/tex]
Hence, the number of design defects were major=5
There are 5 design defects that are also classified as major defects.
To solve this problem, let's break down the information given and organize it in a systematic way using a Venn diagram and algebra.
Given data:
Total number of defects: 28Number of major defects: 14Number of design defects: 8Number of defects that are neither major nor design: 11We need to find out how many of the design defects are also major defects.
Let's denote:
A: Major defectsB: Design defectsA ∩ B: Major design defects (defects that are both major and design)First, note that the defects that are neither major nor design are given as 11.
Therefore, the defects that are either major or design or both are:
[tex]28 - 11 = 17[/tex]
This tells us the total number of defects in sets A or B or both is 17.
Using the principle of inclusion-exclusion, the number of defects that are either major or design can be represented as:
[tex]|A \cup B| = |A| + |B| - |A \cap B|[/tex]
Where:
[tex]|A \cup B| = 17[/tex]
[tex]|A| = 14[/tex]
[tex]|B| = 8[/tex]
Now, substituting the values, we get:
[tex]17 = 14 + 8 - |A \cap B|[/tex]
Solving for [tex]|A \cap B|[/tex]:
[tex]17 = 22 - |A \cap B|[/tex]
[tex]|A \cap B| = 22 - 17 = 5[/tex]
So, the number of defects that are both design defects and major defects is 5.
Consider a bag containing five red marbles, three green ones, one transparent one, three yellow ones, and three orange ones How many possible sets of five marbles are there in which all of them red or green? sets Need Help? Read Tate Tutor
There are 56 possible sets of five marbles where all marbles are either red or green, when selecting from a bag containing five red and three green marbles.
When considering all possible sets of five marbles where each marble is either red or green from a bag containing five red marbles and three green ones, you would use combinations to find the total number of sets.
For the red marbles alone, since there are five red marbles, the number of ways to choose five red marbles is simply C(5, 5), which equals 1. There is only one set which contains all five red marbles because you are choosing all available red marbles.
To find sets containing green marbles, we must consider all combinations of red and green marbles that add up to five. These are C(5, 4) * C(3, 1), C(5, 3) * C(3, 2), and C(5, 2) * C(3, 3). We calculate each of these combinations and then sum them to get the total possible sets that consist of only red and green marbles.
Therefore, the total number of sets composed only of red or green marbles is:
C(5, 5) = 1
C(5, 4) * C(3, 1) = 5 * 3 = 15
C(5, 3) * C(3, 2) = 10 * 3 = 30
C(5, 2) * C(3, 3) = 10 * 1 = 10
Adding these up gives us 1 + 15 + 30 + 10 = 56 possible sets.
There are 3 possible sets of five marbles in which all of them are either red or green.
To solve this problem, we need to consider the different combinations of red and green marbles that can make up a set of five marbles, where all marbles in the set are either red or green.
Firstly, let's consider the case where all five marbles are red. Since there are five red marbles in the bag, there is only one way to choose all five red marbles. This gives us one possible set.
Next, we consider the case where there are four red marbles and one green marble. There are [tex]\(\binom{5}{4}\)[/tex] ways to choose four red marbles from the five available, and [tex]\(\binom{3}{1}\)[/tex]ways to choose one green marble from the three available. Using the combination formula [tex]\(\binom{n}{r} = \frac{n!}{r!(n-r)!}\),[/tex] we calculate:
[tex]\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5 \][/tex]
and
[tex]\[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3}{1} = 3 \][/tex]
Multiplying these together gives us[tex]\(5 \times 3 = 15\)[/tex] possible sets for this case.
Now, let's consider the case where there are three red marbles and two green marbles. There are[tex]\(\binom{5}{3}\)[/tex] ways to choose three red marbles from the five available, and[tex]\(\binom{3}{2}\)[/tex]ways to choose two green marbles from the three available. Calculating the combinations:
[tex]\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \][/tex]
and
[tex]\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3}{1} = 3 \][/tex]
Multiplying these together gives us [tex]\(10 \times 3 = 30\)[/tex] possible sets for this case.
For the case with two red marbles and three green marbles, we have \[tex](\binom{5}{2}\)[/tex] ways to choose two red marbles and [tex]\(\binom{3}{3}\)[/tex] ways to choose three green marbles. Calculating the combinations:
[tex]\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \][/tex]
and
[tex]\[ \binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{1}{1} = 1 \][/tex]
Multiplying these together gives us [tex]\(10 \times 1 = 10\)[/tex] possible sets for this case.
Next, the case with one red marble and four green marbles has [tex]\(\binom{5}{1}\)[/tex]ways to choose one red marble and [tex]\(\binom{3}{4}\)[/tex] ways to choose four green marbles. However, since there are only three green marbles, it is impossible to choose four, so this case is not possible.
Finally, the case with zero red marbles and five green marbles is also not possible because we cannot choose five green marbles from only three available.
Adding up the possible sets from the cases that are possible, we have:
[tex]\[ 1 + 15 + 30 + 10 = 56 \][/tex]
However, upon reviewing the combinations, it is clear that there was an error in the calculations. The correct combinations for the case with three red marbles and two green marbles should be [tex]\(\binom{5}{3} \times \binom{3}{2} = 10 \times 3 = 30\)[/tex], and for the case with two red marbles and three green marbles, it should be [tex]\(\binom{5}{2} \times \binom{3}{3} = 10 \times 1 = 10\)[/tex]. The other cases are correct.
Therefore, the correct total number of possible sets is:
[tex]\[ 1 + 15 + 30 + 10 = 56 \][/tex]
However, the question asks for sets where all marbles are either red or green, and we have incorrectly included sets with both red and green marbles. We need to correct this by only considering the sets with all red marbles or all green marbles.
The correct sets are:
- All five marbles are red: 1 set
- All five marbles are green: This is not possible since there are only three green marbles.
Thus, there is only 1 possible set when considering all marbles must be either red or green. But since the question specifically asks for sets of five marbles, and we can't have a set of five green marbles, we are left with only the set of five red marbles.
The Riverwalk in San Antonio, Texas has several arched pedestrian bridges that cross the San Antonio River. These arches are the arcs of circles. The distance across the river at this bridge is 150 feet, and the height in the center of the arch, above the water is 30 feet. Determine the radius of the circle. Note that the diameter of the circle is not 150 feet.
Answer: radius = 108.75 feet
Hi!
In the drawing you can see the arch that goes from point A to B. The right line from A to B is the length L = 150 feet, across the river. The height is h = 30 feet.
There is a right triangle with hypotenuse R, and the legs are (R-h), and L/2. The Pythagorean theorem says that:
[tex]R^2 = (R-h)^2 + (\frac{L}{2}) ^2 = R^2 -2hR + h^2 + (\frac{L}{2}) ^2\\[/tex]
Then:
[tex]0 = -2hR + h^2 +(\frac{L}{2}) ^2\\R = (h^2 +(\frac{L}{2}) ^2)\frac{1}{2h}[/tex]
Plugging the values of L and h, you get R = 108.75 feet
Answer:
108.75 ft
Step-by-step explanation:
You want the radius of a circular arc that rises 30 ft above a chord of length 150 ft.
CenterThe attached diagram shows the method we used to find the radius. The arc is symmetrical about its centerline, so we only need to find the perpendicular bisector of another chord to determine the location of the center.
Here, we have chosen a coordinate system with the ends of the arc at (±75, 0) and the center at (0, 30). The midpoint of the chord joining the two points on the right will be (75/2, 30/2) = (37.5, 15).
The slope of that chord is ...
[tex]m=\dfrac{\text{rise}}{\text{run}}=\dfrac{-30}{75}=-\dfrac{2}{5}[/tex]
Perpendicular bisectorThe equation of the perpendicular bisector will be the equation of a line with slope -1/m = 5/2 through the point (37.5, 15). In point-slope form, that equation is ...
[tex]y-15=\dfrac{5}{2}(x-37.5)\\\\y=\dfrac{5}{2}x-78.75[/tex]
We note that the y-intercept is -78.75. The distance from that point to the top of the arch is ...
30 -(-78.75) = 108.75
The radius of the circle is 108.75 ft.
__
Additional comment
We can use another relation to solve this problem even more simply. Consider the 150 ft chord divided into two pieces by a diameter that has one segment length 30, and the other equal to the diameter less 30. The products of these segment lengths are the same, so we have ...
75·75 = 30·(d -30)
d-30 = 187.5
d = 217.5 ⇒ r = d/2 = 108.75
The number of species n found on islands typically increases with the area of the island A. Suppose that this relationship is such that the rate of increase with island area is always proportional to the density of species (that is, number of species per unit area) with a proportionality constant k between 0 and 1. Find the function that describes the species-area relationship. (Let n(1) = n1.)
Answer:
[tex]n(A) = n_1A^k[/tex]
Step-by-step explanation:
Taking into account that the growth rate of the number of species on the island is proportional to the density of species (number of species between area of the island), a model based on a differential equation is proposed:
[tex]\frac{dn}{dA} = k\frac{n}{A}[/tex]
This differential equation can be solved by the method of separable variables like this:
[tex]\frac{dn}{n} = k\frac{dA}{A}[/tex] with what you get:
[tex]\int\ {\frac{dn}{n}}\ = k\int\ {\frac{dA}{A}}[/tex]
[tex]ln|n| = kln|A|+C[/tex]. Taking exponentials on both sides of the equation:
[tex]e^{ln|n|} = e^{ln|A|^{k}+C}[/tex]
[tex]n(A) = e^{C}A^{k}[/tex]
how do you have to [tex]n (1) = n_1[/tex], then
[tex]n(A) = n_1A^k[/tex]
The slopeof a linear function h(x) is 2. Suppose the function is translated 8 units up to get d(x). How can h(x) be translated to the left or right to represent the same function d(x)?
Answer:
4 units to the right
Step-by-step explanation:
Since all we know about function h(x) is that it has slope 2, we don't know its y intercept, so let's call it b and write the equation of that first line as:
[tex]h(x) = 2x + b[/tex]
Now, we translated that function up 8 units to get function d(x). This vertical displacement is expressed by adding 8 units to function h(x):
[tex]d(x) = h(x) + 8 = 2x+b+8[/tex]
We want to translate this last function d(x) either to the right or to the left in order to get the same expression as for function h(x).
Recall that translations to the right or left a fix number of units "c" affects the "x" coordinate of the function (by adding "c", we move to the left c units the graph of the function, and by subtracting "c" we move to the right the graph of the function).
So represent a generic shift for d(x) for example to the right in c units:
[tex]d(x) = 2(x-c)+x^{2} +b+8= 2x-2c+b+8[/tex]
Now we want this new shifted function to be EXACTLY as h(x) so we write the equality:
[tex]2x-2c+8+b=h(x)=2x+b\\2x-2c+8+b = 2x+b[/tex]
We can solve in the equation for the value "c" we need for the translation:
[tex]2x-2c+8+b = 2x+b\\-2c+8=0\\2c=8\\c=\frac{8}{2} = 4[/tex]
subtracting 2x from both sides, and also subtracting "b" from both sides.
This means that the translation to the right by 4 units will result on the exact same graph as for function h(x)
NetSell, a the TV remote control supplier for Lumyn Electronics, has a weekly production cost of q TV remote controls that is given by
C(q) = 0.000004q^3 - 0.03q^2 + 100q + 75,000
where q is in the interval [0, 10,000].
The demand function for this product is given by
p(q) = -0.005q + 200.
Based on this information, find the following:
a) The marginal cost for the company.
b) The marginal revenue for the company.
c) The marginal profit for the company when 2,000 and 7,000 TV remote controls are manufactured.
Answer:
a. [tex]\frac{dC(q)}{dq} = 0.000012q^2 -0.06q + 100[/tex]
b. [tex]\frac{dR(q)}{dq}=-0.01q+200[/tex]
c.
[tex]U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152[/tex]
[tex]U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138[/tex]
Step-by-step explanation:
a) The marginal cost function is given by the derivative of the total cost function, in this way the marginal cost function for this company is:
[tex]\frac{dC(q)}{dq} = \frac{dC(q)}{dq} (0.000004q^ 3 - 0.03q ^ 2 + 100q + 75000) = 0.000012q^2 -0.06q + 100[/tex]
b) The income function is given by the relation [tex]R (q) = P (q) q = -0.005q^2 + 200q[/tex].
The marginal revenue function for the company is given by the derivative of the revenue function, in this way the marginal revenue function is:
[tex]\frac{dR(q)}{dq}= -0.01q+200[/tex]
(c) The profit function of the company is given by the relation [tex]U (q) = R (q) - C (q)[/tex], and the marginal utility function is given by the derivative of the utility function, in this way , the marginal utility function is:
[tex]\dfrac{dU(q)}{dq}=R'(q) - C'(q) = -0.01q+200 - (0.000012q^2-0.06q+100) = -0.000012q^2+0.05q+100[/tex]
When q = 2000, the marginal utility is:
[tex]U'(2000)=-0.000012(2000)^2+0.05(2000)+100 = 152[/tex]
When q = 7000, the marginal utility is:
[tex]U'(7000)=-0.000012(7000)^2+0.05(7000)+100 = -138[/tex]
The score on a trivia game is obtained by subtracting number of incorrect answers from twice the number of correct answers. If a player answered 40 questions and obtained a score of 50, how mary questions did the player answer correctly?
Answer:
30
Step-by-step explanation:
I solved this question algebraically. First a variable (I used 'c') is introduced into the equation (for the number of correct answers). Thus we get the equation;
2c - (40 - c) = 50
solving this...
2c - 40 + c = 50
3c = 90
c = 30
So the number of correct answers is 30. (30*2-10=50)
[Alternatively we can use a variable for the number of incorrect answers and get; 2* (40 - a) - a = 50, and solve this equation but this method is longer as you'll need to subtract this answer from 40 to get the number of correct responses.]
What are the odds of winning a game, if the probability of winning the game is 9/10?
Remember:
The probability of an event happening is the fraction: favorable outcomes/total outcomes.
The odds of an event happening treats the event as a win. The odds are given by the ratio: wins: losses
9:1
9:10
1:10
1:9
Final answer:
The odds of winning the game with a probability of 9/10 are 9:1, which means there are 9 chances of winning for every 1 chance of losing.
Explanation:
The question asks for the odds of winning a game given a probability of winning is 9/10. To find the odds in favor of winning, we calculate the ratio of the probability of winning to the probability of not winning. There are 9 chances to win for every 10 chances, so there is 1 chance out of 10 of not winning. Therefore, the odds in favor of winning are 9:1 (9 chances to win versus 1 chance to lose).
The odds of winning the game are 9:1, meaning for every 9 times one wins, one can expect to lose approximately 1 time.
The odds of winning a game, given the probability of winning is 9/10, can be calculated by considering the definition of odds. The odds in favor of an event are given by the ratio of the number of favorable outcomes to the number of unfavorable outcomes.
In this case, the probability of winning is 9/10, which means there are 9 favorable outcomes for every 10 total outcomes (including both wins and losses).
To find the odds, we subtract the probability of winning from 1 to get the probability of losing, which is 1 - 9/10 = 1/10. Now, we have 9 favorable outcomes (wins) and 1 unfavorable outcome (loss) since the total outcomes are 10.
Therefore, the odds of winning the game are 9 favorable outcomes to 1 unfavorable outcome, which is 9:1.
The correct answer is 9:1.
The probability of winning is given as 9/10.
- The probability of losing is the complement of the probability of winning, which is 1 - 9/10 = 1/10.
- Odds are calculated by comparing the number of favorable outcomes to the number of unfavorable outcomes.
- Here, there are 9 favorable outcomes (wins) and 1 unfavorable outcome (loss), so the odds of winning are 9:1.
- The odds format does not consider the total number of outcomes but rather the ratio of wins to losses. Hence, we do not include the total outcomes in the odds ratio.
If aſc and a +b = C, prove that a|b.
Answer:
Assuming a being a divisor of c and [tex]a,b,c \in \mathbb{Z}[/tex]
Step-by-step explanation:
We are told that [tex]a \mid c[/tex]
and that [tex]a+b =c[/tex]
which means that
[tex]\exists k \in \mathbb{N}[/tex] so that [tex]c = k.a[/tex]
so we can rewrite [tex]a+b[/tex] as
[tex]a+b = c = k.a[/tex]
[tex]b = k.a-a = (k-1).a[/tex]
and as [tex]k \in \mathbb{N}[/tex]
We have that either
Showing that [tex]a \mid c[/tex]
Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T − 0.02t 1 8.50, where T is temperature in °C and t represents years since 1900. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100.
Answer:
a) The slope represents the change per year in the earth's surface temperature (0.02 ºC) and the T-intercept represents the average earth's surface temperature in 1900 (18.5ºC)
b) The average global surface temperature will be 22.5ºC in 2100.
Step-by-step explanation:
I think there's a "+" sign missing in the function you wrote, and that you meant to write T= 0.02t + 18.50 and I'll solve the problem for this function.
If I am wrong with the values or the + you can always substitute the values you have in your function based on the procedure I'll write down.
So, this function is T= 0.02t + 18.50.
A linear function is expressed as y = f(x) = mx + b where m is the slope of the function and b is the interception with the y-axis.
a) In this particular function, we would have that the slope of the function is 0.02 and it represents the change per year in the earth's surface temperature, since T is temperature in ºC, the change per year would be of 0.02ºC.
On the other hand, the T-intercept would be the value of the function when t=0
T(0) = 0.02(0) +18.50
T(0) = 18.50
This 18.50 represents the average surface temperature of the earth when measures started, meaning, the average surface temperature of the earth in 1900 was 18.50
b) To predict the average global surface temperature in 2100, we are going to substitute 200 in the function (since 2100 - 1900 = 200)
T(200) = 0.02 (200) + 18.50 = 4 + 18.50 = 22.5
Therefore, the average global surface temperature in 2100 will be 22.5ºC
Final answer:
The slope of the linear function represents the rate of temperature increase per year while the T-intercept represents the estimated global surface temperature in 1900. Substituting t=200 into the equation predicts an average global surface temperature of 12.50°C in 2100.
Explanation:
Understanding the Linear Model of Earth's Temperature Rise
The linear function given is T = 0.02t + 8.50, where T is the temperature in degrees Celsius and t represents the years since 1900. The slope of this function is 0.02, which indicates the average rate at which the average global surface temperature is increasing per year. The T-intercept is 8.50°C, which can be interpreted as the estimated global surface temperature in the base year, 1900.
To project the average global surface temperature in 2100, we simply substitute t = 200 into the equation (since 2100 is 200 years after 1900) and calculate:
T = 0.02(200) + 8.50 = 4 + 8.50 = 12.50°CTherefore, using this model, the predicted average global surface temperature in 2100 is 12.50°C.
Assume that men's weights are normally distributed with a mean of 172 lb and a standard deviation of 29 lb. If 4 man is randomly selected, find the probability that they have a mean weight between 160 lb and 180 lb.(up to four decimal place, please)
The probability that the mean weight of 4 randomly selected men is between 160 lb and 180 lb is approximately 0.5025 (up to four decimal places).
To find the probability that the mean weight of 4 randomly selected men falls between 160 lb and 180 lb, you can use the Central Limit Theorem.
According to the Central Limit Theorem, the distribution of sample means will be approximately normally distributed, regardless of the distribution of the population, as long as the sample size is sufficiently large.
The formula for the standard error of the mean [tex](\(SE\))[/tex] is given by:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
where:
- [tex]\(\sigma\)[/tex] is the population standard deviation,
- [tex]\(n\)[/tex] is the sample size.
In this case, [tex]\(\sigma = 29\)[/tex] lb (population standard deviation) and [tex]\(n = 4\)[/tex] (sample size).
[tex]\[ SE = \frac{29}{\sqrt{4}} = \frac{29}{2} = 14.5 \][/tex]
Now, you can find the z-scores for the values 160 lb and 180 lb using the formula:
[tex]\[ z = \frac{X - \mu}{SE} \][/tex]
where:
- [tex]\(X\)[/tex] is the value you're interested in,
- [tex]\(\mu\)[/tex] is the population mean,
- [tex]\(SE\)[/tex] is the standard error of the mean.
For [tex]\(X = 160\)[/tex] lb:
[tex]\[ z_{160} = \frac{160 - 172}{14.5} \approx -0.8276 \][/tex]
For [tex]\(X = 180\)[/tex] lb:
[tex]\[ z_{180} = \frac{180 - 172}{14.5} \approx 0.5517 \][/tex]
Now, you can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
The probability that the mean weight is between 160 lb and 180 lb is given by the difference in these probabilities:
[tex]\[ P(160 < \text{mean} < 180) = P(z_{180}) - P(z_{160}) \][/tex]
Substitute the z-scores:
[tex]\[ P(160 < \text{mean} < 180) = P(0.5517) - P(-0.8276) \][/tex]
Using a standard normal distribution table or calculator, you can find these probabilities. Once you find the probabilities, subtract the smaller from the larger to get the probability in the specified range.
[tex]\[ P(160 < \text{mean} < 180) \approx 0.7082 - 0.2057 \][/tex]
[tex]\[ P(160 < \text{mean} < 180) \approx 0.5025 \][/tex]
So, the probability that the mean weight of 4 randomly selected men is between 160 lb and 180 lb is approximately 0.5025 (up to four decimal places).
Rita Crudwell was the appointed comptroller and treasurer of the town of Dixon, Illinois (population 15,700 in 2010) for 29 years. Despite her busy, dual job, she was able to establish and run a world renowned quarter horse breeding operation, amassing 52 world championships. I do now know how she found the time and energy to do this but I do know where she got the money: over the course of her last 22 years in office, she embezzled $53,740,394 million dollars from the town (caught in 2012, she is currently serving a 19 year prison term) What would be the "hourly wage"
dollars from the town (caught in 2012, she is currently serving a 19 year prison term) What would be the "hourly wage" for a person that wants to make $53,740,394 million over exactly 22 years? (NOTE: assume 1 year 365.256 days) 1-2 If a certain physics professor gets $65,000/year, how many years would it take for such a person to make $53,740,394 million?
Answer:
a) $278.65 million per hour
b) 826.77 million years
Step-by-step explanation:
Given:
Total money = $53,740,394 million
Time taken to earn the money = 22 years
Now,
1 year = 365.256 days
and 1 day = 24 hours
thus,
22 years = 22 × 365.256 × 24 = 192855.168 hours
Therefore,
for a person that wants to make $53,740,394 million over exactly 22 years, will have hourly wage
= [tex]\frac{\textup{money to be earned}}{\textup{Time taken in hours}}[/tex]
= [tex]\frac{\textup{53,740,394 million}}{\textup{192855.168 hours}}[/tex]
= $278.65 million per hour
Now,
Salary of a physics professor = 65000 /year
Thus,
for physics professor to earn $53,740,394 million number of years taken will be
= [tex]\frac{\textup{$53,740,394 million}}{\textup{65000 /year}}[/tex]
= 826.77 million years
Final answer:
To earn $53,740,394 through embezzlement over 22 years equates to an "hourly wage" of $1,174.60, assuming a 40-hour workweek. It would take a physics professor making $65,000 a year approximately 827 years to earn the same amount.
Explanation:
Calculating the "hourly wage" for embezzling $53,740,394 over 22 years requires us to first determine the total hours worked over those years, assuming a standard full-time job framework of 40 hours per week. Also, to calculate how many years it would take a physics professor earning $65,000 a year to amass the same amount of money embezzled requires straightforward division.
Step 1: Calculate the Total Number of Hours Worked Over 22 years
Number of weeks in a year = 52
Hours worked per week = 40
Total hours worked in a year = 52 weeks * 40 hours/week = 2,080 hours
Total hours worked over 22 years = 2,080 hours/year * 22 years = 45,760 hours
Step 2: Calculate the Hourly "Wage" from Embezzlement
Total embezzled = $53,740,394
Hourly "wage" = Total embezzled / Total hours worked over 22 years = $53,740,394 / 45,760 hours = $1,174.60 per hour
Step 3: Calculate the Years for a Physics Professor to Earn $53,740,394
Annual salary = $65,000
Years required = Total embezzled / Annual salary = $53,740,394 / $65,000 ≈ 826.78 years
There are 16 tablespoons in one cup. Which table correctly relates the number of cups to the number of tablespoons.
Answer:
Step-by-step explanation:
The first one
1 16
2 32
4 64
8 128
The table that correctly relates the number of cups to the number of tablespoons is the first table.
How to interpret integral multiplication?Suppose that there are two positive integer numbers( numbers like 1,2,3,.. ) as a and b
Then, their multiplication can be interpreted as:
[tex]a \times b = a + a + ... + a \: \text{(b times)}\\\\a \times b = b + b +... + b \: \text{(a times)}[/tex]
For example,
[tex]5 \times 2 = 10 = 2 + 2 + 2 + 2 + 2 \: \text{(Added 2 five times)}\\or\\5 \times 2 = 10 = 5 + 5 \: \text{(Added 5 two times)}[/tex]
It is specified that:
In 1 cup, there are 16 tablespoons
In 2 cups, there would be 16+16 tablespoons,
and so on,
in 'n' cups, there would be 16+16+...+16 (n times ) = 16 × n tablespoons.
Checking all the tables one by one:
Second table is incorrect because it says that 16 cups = 1 tablespoon which is wrong.
The third table is saying in 32 cups there are only 16 tablespoons, which is obviously wrong as number of tablespoon is always going to be bigger than the number of cups.
The first table is correct since:
Number of cups(n) Number of tablespoons ( 16 × n)
1 16 × 1 =16
2 16 × 2 = 32
4 16 × 3 = 64
8 16 × 4 = 128
So it follows the formula we obtained for the number of tablespoons for given number of cups.
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Which of the following is the best example of a business economic decision.
Answer:
C. Hiring part-time employees to save on costs
Step-by-step explanation:
It is good economically to not have to pay the employee full time but still have someone to help the business so they are there a shorter period of time
Though this question is not mathematics, the best example of a business economic decision is C) Hiring part-time employees to save on costs.
What is an economic decision?Economic decisions revolve around the following economic activities in the use of economic resources:
SavingExchangeInvestmentProductionDistributionConsumption.Business economic decisions are made to achieve business goals, including savings, efficiency, and effectiveness.
Thus, the best example is of a business economic decision is Option C because it involves the achievement of a business goal, savings.
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The average summer vacation costs $2252. If 82% of this amount is charged on credit cards, then what amount of the vacation cost is charged?
Answer: $1846.64
Step-by-step explanation:
Given : The average summer vacation costs $2252.
If 82% of this amount is charged on credit cards, then the amount of the vacation cost is charged will be :-
[tex]82\%\text{ of }\$2252\\\\\text{Convert percent into fraction by dividing it by 100, we get}\\\\=\$[\dfrac{82}{100}\times2252]\\\\=\$[\dfrac{184664}{100}]\\\\=\$1846.64[/tex]
Therefore the amount of the vacation cost is charged= $1846.64
14 of 16 (12 complete) In a baseball league consisting of 7 teams, each team plays each of the other teams twice. How many games will be played? There will be games played.
Answer: There will be 42 games played.
Step-by-step explanation:
The number of combination to arrange n things if r things are taken at a time:-
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]
Similarly, the number of combination to arrange 7 teams if 2 teams are taken at a time:-
[tex]^7C_2=\dfrac{7!}{2!(7-2)!}\\\\=\dfrac{7\times6\times5!}{2\times5!}=21[/tex]
∴ Number of combination to arrange 7 teams if 2 teams are taken at a time=21
Also, each team plays each of the other teams twice.
Then, the number of games will be played: [tex]21\times2=42[/tex]
Hence, There will be 42 games played.
Final answer:
To calculate the total number of games played in a league with 7 teams where each team plays the others twice, use the formula n(n - 1) × 2, which results in 7 * 6 * 2 = 84 games.
Explanation:
To find out the total number of games played in a baseball league with 7 teams where each team plays each of the other teams twice, we can use the formula for the number of games in a round-robin tournament: n(n - 1), where n is the number of teams. Since each team plays each other twice, the formula modifies to n(n - 1) × 2.
Substituting the number of teams (7) into the formula gives us: 7 * (7 - 1) × 2 = 7 * 6 × 2 = 42 × 2 = 84 games in total.
250 ml of a liquid medication contains 125 mg of Drug Z. If the patient gets an injection of 50 ml of this medication, how much drug Z is in that injection
Final answer:
To find the amount of Drug Z in the 50 ml injection, set up a proportion based on the known concentration (125 mg in 250 ml). Solving for the unknown quantity gives 25 mg of Drug Z in the injection.
Explanation:
To determine how much of Drug Z is in a 50 ml injection when 250 ml of the medication contains 125 mg of Drug Z, you can use a proportion. The proportion can be set up as follows:
(125 mg of Drug Z) / (250 ml of medication) = (X mg of Drug Z) / (50 ml of medication)
To solve for X, which represents the amount of Drug Z in the 50 ml injection, you cross-multiply and divide:
(125 mg) × (50 ml) = (250 ml) × (X mg)
6250 mg·ml = 250 ml·X mg
Divide both sides by 250 ml to isolate X:
X = (6250 mg·ml) / (250 ml)
X = 25 mg of Drug Z
Therefore, the 50 ml injection contains 25 mg of Drug Z.
Assume that a customer shops at a local grocery store spending ak average if $200 a week, resulting in the retailer earning a $25 profit each week from this customer. Assuming the shopper visits the store all 52 weeks of the year, calculate the customer lifetime value if this shopper remains logal over a 10-year life-span.
Answer:
$13,000
Step-by-step explanation:
We have been given that the retailer earns a $25 profit each week from a customer.
The shopper visits the store all 52 weeks of the year, so number of weeks in 10 years would be 10 times 52 that is 520 weeks.
We know that the customer lifetime value stands for a prediction of the net profit attributed to the entire future relationship with a customer.
The customer lifetime value would be profit made per week times number of weeks the customer will shop.
[tex]\text{The customer lifetime value}=\$25\times 520[/tex]
[tex]\text{The customer lifetime value}=\$13,000[/tex]
Therefore, the customer lifetime value would be $13,000.
combinatorics: what is the coefficient of (a^2)(b^3)(c) in (2a - b + 3c)^6?
Hence, the coefficient of a²b³c = -720.
Step-by-step explanation:
As from the question,
The general formula to find the coefficient is given by Binomial theorem:
That is, the coefficient of [tex]x^{\alpha}\cdot y^{\beta}\cdot z^{\gamma}[/tex] in (x + y + z)ⁿ is given by:
[tex]\frac{n!}{\alpha ! \cdot \beta ! \cdot \gamma !} (x)^{\alpha} \cdot (y)^{\beta} \cdot (z)^{\gamma}[/tex]
Now,
From the question we have
[tex](2a-b+3c)^{6}[/tex] having n = 6
∴ x = 2a
y = -b
z = 3c
Now,
The coefficient of a²b³c, that is
α = 2
β = 3
γ = 1
Therefore the coefficient of a²b³c =
[tex]= \frac{6!}{2 ! \cdot 3 ! \cdot 1 !} (2a)^{2} \cdot (-b)^{3} \cdot (3c)^{1}[/tex]
[tex]= \frac{6!}{2 ! \cdot 3 ! \cdot 1 !} 4(a)^{2} \cdot (-b)^{3} \cdot (3c)[/tex]
= -720 a²b³c
Hence, the coefficient of a²b³c = -720.
a box of 100 nails weighs 1 5/8 pounds. mark used 3 1/3 boxes of naila to build a 2 story tree house. how many pounds of nails did he use?
Answer:
5 5/12 pounds of nails
Step-by-step explanation:
(3 1/3 boxes) × (1 5/8 pounds/box) = (10/3)(13/8) pounds = 65/12 pounds
= 5 5/12 pounds
Mark used approximately 5.41 pounds of nails to build the tree house. This is found by converting the given fractions to decimal and then multiplying the weight of one box of nails by the number of boxes use
Explanation:In the problem, we know that a box of 100 nails weighs 1 5/8 pounds. We are also told that Mark used 3 1/3 boxes of nails. Therefore, we first need to find out how much one box of nails weighs, and then multiply that weight by the number of boxes Mark used.
To convert 1 5/8 to a decimal, we know that 5/8 = 0.625, so 1 5/8 = 1.625 pounds. Mark used 3 1/3 boxes, and 1/3 converted to a decimal is approximately 0.33, so Mark used approximately 3.33 boxes. By multiplying 1.625 pounds by 3.33, we find that he used approximately 5.41 pounds of nails to build the tree house.
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Is the set \mathbb{Z} a group under the operation a*b = a - b + ab ? Justify your answer.
Answer:
Step-by-step explanation:
Given that *is the binary operation in the sets of integers.
[tex]a*b = a - b + ab[/tex]
closure: a-b+ab is again an integer belongs to Z. Hence closure is true.
Associativity: [tex]a*(b*c) = a*(b+c-bc)\\= a-b-c+bc+ab+ac-abc\\= a-b-c +ab+bc+ca-abc[/tex]
[tex](a*b)*c=(a+b-ab)*c\\=a+b-ab-c+ac+bc-abc\\[/tex]
The two are not equal. Hence this cannot be a group as associtiavity does not hold good.
Translate this formula into an english sentence using this "dictionary":
A= Salt is added to the soltuiton
D= the solution's boiling point drops
M= The manuel is correct
S= The solution boils sooner
(A --> D) --> (A --> S)
Answer: If the solution's boiling point drops because salt is added to the soltuiton, then the reason the solution boils sooner is the addition of salt to the solution
Step-by-step explanation:
The re are two cause - consequence relations inside an other one.
I use 3 different cause - consequence connectors not to mix them up.
A --> D equals “ D because A “
A --> S equals “ the reason S happens is A “
(A --> D) --> (A --> S) Equals “ if (A --> D) then (A --> S) “
A 10-mL graduate weighs 42.745 g. When 5 mL of distilled water are measured in it, the combined weight of graduate and water is 47.675 g. By definition, 5 mL of water should weigh 5 g. Calculate the weight of the measured water and express any deviation from 5 g as a percentage of error.
Answer:
The measured weight of the water = 4.93 g
The percentage error in measured value = 1.4%
Step-by-step explanation:
Given:
Weight of the 10-mL graduate = 42.745 g
Combined weight of graduate and 5 mL water = 47.675 g
5 mL of water should weight = 5 g
thus, actual value of 5 mL water = 5 g
Now,
The measured weight of the water
= (Combined weight of graduate + 5 mL water) - Weight of the 10-mL graduate
= 47.675 g - 42.745 g
= 4.93 g
The percentage error in measured value
= [tex]\frac{\textup{Actual value-calculated value}}{\textup{Actual value}}\times100[/tex]
= [tex]\frac{\textup{5-4.93}}{\textup{5}}\times100[/tex]
or
= 1.4%
The weight of the measured water is 4.930 g, which is slightly less than the theoretical 5 g. The percentage error of the measurement is 1.4%, falling within the acceptable ±5% error range.
Explanation:To calculate the weight of the measured water and the percentage error, we use the information provided. The weight of the empty graduate is 42.745 g, and the combined weight of the graduate and water is 47.675 g. Subtracting the weight of the empty graduate from the combined weight gives us the weight of the water only:
Weight of water only = 47.675 g - 42.745 g = 4.930 g
Since 5 mL of distilled water should ideally weigh 5 g, we can then calculate the percentage error:
Percentage error = [(Target amount of water - Actual volume dispensed) / Target amount of water] × 100%
Percentage error = [(5 g - 4.930 g) / 5 g] × 100%
Percentage error = [0.07 g / 5 g] × 100%
Percentage error = 1.4%
The measured water weighs slightly less than the expected 5 g, with a percentage error of 1.4%, which is within the acceptable range of ±5% error.
Is the statement given below true or false? Give
ajustification if true or a counterexample if false
ALL ISOSCELES TRIANGLES WITH CONGRUENT VERTEX ANGLES
ARESIMILAR.
Answer:
Yes, "All isosceles triangles with congruent vertex angles are similar".
Step-by-step explanation:
Consider the provided statement.
All isosceles triangles with congruent vertex angles are similar.
As we know that the two sides of an isosceles triangle are same.
It is given that the isosceles triangles with congruent vertex angles.
If vertex angles are congruent it means the opposite side of those angles are congruent. Also the sums of the base angles are the same,
As we know the base angles of an isosceles triangle are congruent, so by the AAA similarity we can say "All isosceles triangles with congruent vertex angles are similar".
Or
let say ΔABC and ΔDEF are isosceles triangles where AB=DE and AC=DF
It means ∠B=∠E and ∠C=∠F also it is given that ∠A=∠D
Thus. from AAA similarity we can say "All isosceles triangles with congruent vertex angles are similar".
A town's population has been growing linearly. In 2003, the population was 59.000 and the population has been growing by 1,700 people each year. a. Write a formula for the population x years after 2003. b. What will the town's population be in 2007? c. In what year will the population be 77,700 people?
Answer:
In 2003, the population was 59000 and the population has been growing by 1,700 people each year.
A.
The equation will be:
59000+1700x = (population 'x' years after 2003)
For x, you plug in the amount of years after 2003.
Like if it is the year 2003, the population is [tex]59000+1700(0)[/tex]
= 59000
when it is year 2005, the population is [tex]59000+1700(2)[/tex]
= 62400
B.
The town's population in 2007 will be :
[tex](2007-2003=4)[/tex]
[tex]59000+1700(4)[/tex]
Population = 65800
C.
[tex]59000+1700x=77700[/tex]
=> [tex]1700x=18700[/tex]
x = 11
Means [tex]2003+11=2014[/tex]
Hence, by year 2014 the population will be 77700.
What is the cost of constructing a fence 6 feet, 6 inches, high around a lot measuring 90 feet by 175 feet, if the cost of erecting the fence is $1.25 per linear foot and the cost of materials is $0.825 per square foot of fence? Round your answer to the nearest cent.
A. $1,752.62 B. $2,054.63 C. $2,084.62 D. $3,504.63
Answer:
The total cost is $3504.625
Step-by-step explanation:
1 inch = 0.0833333 feet
Height = 6 feet 6 inches =[tex]6+6 \times 0.0833333=6.5 feet[/tex]
Length = 90 feet
Breadth = 175 feet
Perimeter = [tex]2(l+b)=2(90+175) = 530 feet[/tex]
Cost of erecting the fence is $1.25 per linear foot .
So, Cost of erecting the fence of 530 feet = [tex]530 \times 1.25[/tex]
Cost of erecting the fence of 530 feet = [tex]662.5[/tex]
Area of fence = [tex]2(l+b)h=530 \times 6.5=3445 feet^2[/tex]
Cost of materials is $0.825 per square foot of fence
Cost of materials of 3445 square foot of fence = [tex]3445 \times 0.825[/tex]
= [tex]2842.125[/tex]
Total cost = $2842.125+$662.5
Total cost = $3504.625
Hence the total cost is $3504.625
Find the critical value from the Studentized range distribution for H0: μ1 = μ2 = μ3 = μ4 = μ5, with n = 35 at α = 0.01. Provide answer to three decimal places (example, 3.254).
Answer:
t-value = 2.441
Step-by-step explanation:
Let's assume that this is a one-tailed test to calculate the critical value, the process is this:
Calculate alpha (α): α = 1 - (confidence level / 100) , but we already have this α=0.01Find the critical probability (p*): p* = 1 - α/2 = 1-0.005=0.995Then, the critical value would be shown as a t statistic, but for this we need:degrees of freedom (df)= n-1=35-1=34
The critical t statistic (t*) or critical value is the t-value having degrees of freedom equal to 34 and a cumulative probability equal to 0.995.From the attached table we can see:
t-value = 2.441
Final answer:
The critical value from the Studentized range distribution for the given hypothesis test can be found using the qtukey() function in R or by looking it up in the table. For n=35 and α=0.01, the critical value is approximately 4.116.
Explanation:
To find the critical value from the Studentized range distribution, we need to use the qtukey() function in R or look up the value in the Studentized range distribution table.
Since n=35 and α=0.01, we have k=5 (number of groups) and df=5*(n-1)=170.
Using R, the critical value for a 5% significance level (α=0.05) is approximately 4.116.
Samples of a cast aluminum part are classified on the basis of surface finish and edge finish. The results of 100 parts are summarized as follows. Edge finish Excellent Good Surface Excellent 75 4Finish Good 15 6Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent edge finish. If a part is selected at random, determine the following probabilities: (a) P(A B), (b) P(A U B) and (c) P(A' U B)
Answer:
[tex]a. \hspace{3} P(A\bigcap B) = \frac{3}{4}\\\\b. \hspace{3} P(A\bigcup B) = \frac{47}{50}\\\\c. \hspace{3} P(A'\bigcup B) = \frac{17}{20}\\\\[/tex]
Step-by-step explanation:
The information is configured in a double entry table in which the finishing information for the edge and surface is recorded, thus:
[tex]\begin{array}{cccc}&E&B&Total\\E&75&4&79\\B&15&6&21\\&90&10&100\\\end{array}[/tex]
Let A denote the event that a sample has excellent surface finish, and let B denote the event that a sample has excellent edge finish.
[tex]a. \hspace{3} P(A\bigcap B) = \frac{75}{100} = \frac{3}{4}\\\\b. \hspace{3} P(A\bigcup B) = P(A) + P(B) - P(A\bigcap B) = \frac{90}{100}+\frac{79}{100} - \frac{75}{100} = \frac{94}{100} =\frac{47}{50}\\\\c. \hspace{3} P(A'\bigcup B) = P(A') + P(B) - P(A'\bigcap B) = \frac{10}{100}+\frac{79}{100} - \frac{4}{100} = \frac{85}{100} =\frac{17}{20}\\\\[/tex]
How many grams of digoxin (Lanoxin) would be required to make 25,000 tablets each containing 250 mcg of digoxin?
Answer:
6.25 grams would be required
Step-by-step explanation:
This problem can be solved as a rule of three problem.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
First step: The first step is determining how many mcg are used to make 25,000 tablets.
The problem states that each tablet contains 250 mcg of digoxin. So:
1 tablet - 250mcg
25,000 tables - x mcg
x = 25,000*250
x = 6,250,000 mcg
25,000 tables have 6,250,000mcg
Final step: Conversion of 6,250,000mcg to g
Each g has 1,000,000 mcg. How many g are in 6,250,000mcg? So:
1g - 1,000,000 mcg
xg - 6,250,000 mcg
1,000,000x = 6,250,000
[tex]x = \frac{6,250,000}{1,000,000}[/tex]
x = 6.25g
6.25 grams would be required