Answer:
91 people take Russian
26 people take French and Russian but not German
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the students that take French.
-The set B represents the students that take German
-The set C represents the students that take Russian.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of students that take only Franch, A \cap B is the number of students that take both French and German , A \cap C is the number of students that take both French and Russian and A \cap B \cap C is the number of students that take French, German and Russian.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There are 155 people in my school. This means that:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 155[/tex]
The problem states that:
90 take Franch, so:
[tex]A = 90[/tex]
83 take German, so:
[tex]B = 83[/tex]
22 take French, Russian, and German, so:
[tex]A \cap B \cap C = 22[/tex]
42 take French and German, so:
[tex]A \cap B = 42 - (A \cap B \cap C) = 42 - 22 = 20[/tex]
41 take German and Russian, so:
[tex]B \cap C = 41 - (A \cap B \cap C) = 41 - 22 = 19[/tex]
22 take French as their only foreign language, so:
[tex]a = 22[/tex]
Solution:
(1) How many take Russian?
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + 19 + 22[/tex]
[tex]C = c + (A \cap C) + 41[/tex]
First we need to find [tex]A \cap C[/tex], that is the number of students that take French and Russian but not German. For this, we have to go to the following equation:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
[tex]90 = 22 + 20 + (A \cap C) + 22[/tex]
[tex](A \cap C) + 64 = 90[/tex].
[tex](A \cap C) = 26[/tex]
----------------------------
The number of students that take Russian is:
[tex]C = c + 26 + 41[/tex]
[tex]C = c + 67[/tex]
------------------------------
Now we have to find c, that we can find in the equation that sums all the subsets:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 155[/tex]
[tex]22 + b + c + 20 + 26 + 19 + 22 = 155[/tex]
[tex]b + c + 109= 155[/tex]
[tex]b + c = 46[/tex]
For this, we have to find b, that is the number of students that take only German. Then we go to this eqaution:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 19 + 20 + 22[/tex]
[tex]B = b + 61[/tex]
[tex]b + 61 = 83[/tex]
[tex]b = 22[/tex]
-------
[tex]b + c = 46[/tex]
[tex]c = 46 - b[/tex]
[tex]c = 24[/tex]
The number of people that take Russian is:
[tex]C = c + 67[/tex]
[tex]C = 24 + 67[/tex]
[tex]C = 91[/tex]
91 people take Russian
(2) How many take French and Russian but not German?
[tex](A \cap C) = 26[/tex]
26 people take French and Russian but not German
An eight-sided die, which may or may not be a fair die, has four colors on it; you have been tossing the die for an hour and have recorded the color rolled for each toss. What is the probability you will roll a yellow on your next toss of the die? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
Answer:
Case 1
None of the sides is yellow.
Probability of rolling a yellow on your next toss: zero %
Case 2
All the sides are yellow.
Probability of rolling a yellow on your next toss: 100 %
Case 3
At least one of the sides is yellow.
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of tosses in one hour.
q = number of tosses you rolled a yellow.
Step-by-step explanation:
Case 1
None of the sides is yellow.
Probability of rolling a yellow on your next toss: zero %
Case 2
All the sides are yellow.
Probability of rolling a yellow on your next toss: 100 %
Case 3
At least one of the sides is yellow.
As you do not know if the die is fair or not, the only way to approximate a probability of rolling a yellow is by making a table of frequencies and record the times you have rolled yellow.
If the number of tosses made in an hour is big enough as to draw a conclusion, then according to the Law of Large Numbers, the probability of rolling a yellow in one toss of the die should be
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of tosses
q = number of tosses you rolled a yellow.
Now, suppose you want to roll the dice once more.
As the event of rolling a die is independent of the previous tosses, this means that the probable outcome of the event does not depend on the previous results. So, the probability remains the same.
Probability of tossing a yellow on your next toss of the die
[tex]\frac{q}{N}\times100\%[/tex]
where
N = total number of previous tosses
q = number of tosses you rolled a yellow.
Compare the values of the underlined digits 631,485 and 682 the underlined is 6.
Answer:
First one: Its place value is 600,000.
Second one: Its place value is 600.
Step-by-step explanation:
The fraction of defective integrated circuits produced in a photolithography process is being studied. A random sample of 300 circuits is tested, revealing 17 defectives. (a) Calculate a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool. Round the answers to 4 decimal places. less-than-or-equal-to p less-than-or-equal-to (b) Calculate a 95% upper confidence bound on the fraction of defective circuits. Round the answer to 4 decimal places. p less-than-or-equal-to
Answer:
(a) The confidence interval is: 0.0304 ≤ π ≤ 0.0830.
(b) Upper confidence bound = 0.0787
Step-by-step explanation:
(a) The confidence interval for p (proportion) can be calculated as
[tex]p \pm z*\sigma_{p}[/tex]
[tex]\sigma=\sqrt{\frac{\pi*(1-\pi)}{N} }\approx\sqrt{\frac{p(1-p)}{N} }[/tex]
NOTE: π is the proportion ot the population, but it is unknown. It can be estimated as p.
[tex]p=17/300=0.0567\\\\\sigma=\sqrt{\frac{p(1-p)}{N} }=\sqrt{\frac{0.0567(1-0.0567)}{300} }=0.0134[/tex]
For a 95% two-sided confidence interval, z=±1.96, so
[tex]\\LL = p-z*\sigma=0.0567 - (1.96)(0.0134) = 0.0304\\UL =p+z*\sigma= 0.0567 + (1.96)(0.0134) = 0.0830\\\\[/tex]
The confidence interval is: 0.0304 ≤ π ≤ 0.0830.
(b) The confidence interval now has only an upper limit, so z is now 1.64.
[tex]UL =p+z*\sigma= 0.0567 + (1.64)(0.0134) = 0.0787[/tex]
The confidence interval is: -∞ ≤ π ≤ 0.0787.
Final answer:
To calculate a confidence interval for the fraction of defective circuits, use the formula for the confidence interval of a proportion. The 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.
Explanation:
To calculate a confidence interval for the fraction of defective circuits, we can use the formula for the confidence interval of a proportion. Let p-hat be the proportion of defectives in the sample, which is equal to 17/300 = 0.0567. We can calculate the standard error as [tex]\sqrt{((p-hat*(1-p-hat))/n)[/tex], where n is the sample size.
Using a 95% confidence level, we can find the critical value from the standard normal distribution, which is approximately 1.96. The lower bound of the confidence interval is given by p-hat - z*(standard error), and the upper bound is given by p-hat + z*(standard error).
Lower bound: 0.0567 - [tex]1.96*(\sqrt{((0.0567*(1-0.0567))/300))[/tex] = 0.0182Upper bound: 0.0567 + [tex]1.96*(\sqrt{((0.0567*(1-0.0567))/300))[/tex]= 0.0951Therefore, the 95% two-sided confidence interval for the fraction of defective circuits is 0.0182 ≤ p ≤ 0.0951. The 95% upper confidence bound on the fraction of defective circuits is 0.0951.
9 + 22 = x + y
x = 5
y = ?
Answer:
y = 26
Step-by-step explanation:
9 + 22 = x + y
9 + 22 = 31
31 = x + y
x = 5
31 = 5 + y
31 - 5 = 26
y = 26
Hey!
-----------------------------------------------
Solution:
9 + 22 = 5 + y
31 = 5 + y
31 - y = 5 + y - y
31 - y = 5
31 - y - 31 = 31 - 5
y = 26
-----------------------------------------------
Answer:
y = 26
-----------------------------------------------
Hope This Helped! Good Luck!
The brain volumes (cm3) of 20 brains have a mean of 1094.9 cm3 and a standard deviation of 128.9 cm3. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1392.7 cm3 be significantly high?
Answer:
Yes, a brain volume of [tex] 1392.7 cm^{3}[/tex] is significantly high.
Step-by-step explanation:
Given
Standard deviation = [tex] 128.9 cm^{3}[/tex]Mean = [tex] 1094.9 cm^{3}[/tex]The range rule of thumb indicates that the usual values are within 2 standard deviations from the mean:
minimum usual value = mean - 2 * standard deviation = [tex]1094.9 - 2*128.9=837.1 cm^{3}[/tex]
maximum usual value = mean + 2 * standard deviation = [tex]1094.9 + 2*128.9=1352.7 cm^{3}[/tex]
We can see that [tex] 1392.7 cm^{3}[/tex] is not between [tex] 837.1 cm^{3}[/tex] and [tex] 1352.7 cm^{3}[/tex], which indicates that this value is unusually high.
Using the range rule of thumb, it is found that:
The limit separating the values that are significantly low is 837.1 cm³.The limit separating the values that are significantly high is 1352.7 cm³.1392.7 cm³ is above 1352.7 cm³, thus, a brain volume of 1392.7 cm³ would be considered significantly high.----------------------------
The range rule of thumb states that:
Measures that are more than 2 standard deviations below the mean are significantly low.Measures that are more than 2 standard deviations above the mean are significantly high.----------------------------
The mean is 1094.9.The standard deviation is 128.9.----------------------------
2 standard deviations below the mean is given by:[tex]1094.9 - 2(128.9) = 837.1[/tex]
The limit separating the values that are significantly low is 837.1 cm³.----------------------------
2 standard deviations above the mean is given by:[tex]1094.9 + 2(128.9) = 1352.7[/tex]
The limit separating the values that are significantly high is 1352.7 cm³.----------------------------
1392.7 cm³ is above 1352.7 cm³, thus, a brain volume of 1392.7 cm³ would be considered significantly high.A similar problem is given at https://brainly.com/question/24126815
A ferry leaves Nanaimo to make the 22 km trip to Vancouver at the same time as a forry leaves Vancouver for Nanaimo. The ferry leaving Nanaimo travels 2 km/h faster than the other ferry How far are they from Vancouver when they meet 45 minutes later?
Answer:
They are 11.7475km away from Vancouver when they meet.
Step-by-step explanation:
The first step to solve this problem is modeling the position of each ferry. The position can be modeled by a first order equation in the following format:
[tex]S(t) = S_{0} + vt[/tex], in which [tex]S_{0}[/tex] is the initial position of the ferry, t is the time in hours and v is the speed in km/h.
I am going to say that the positive direction is from Nanaimo to Vancouver, and that Nanaimo is the position 0 and Vancouver the position 22.
First ferry:
Leaves Nanaimo, so [tex]S_{0} = 0[/tex]. It is 2km/h faster than the second ferry, so i am going to say that [tex]v = v + 2[/tex]. It moves in the positive direction, so v is positive. The equation of the position of this train is modeled as:
[tex]S_{1}(t) = 0 + (v+2)t[/tex],
Second ferry:
Leaves Vancouver, so [tex]S_{0} = 22[/tex]. It has a speed of v, that is negative, since it moves in the negative direction. So
[tex]S_{2}(t) = 22 - vt[/tex]
The problem states that they meet in 45 minutes. Here we have to pay attention. Since the speed is in km/h, the time needs to be in h. So 45 minutes = 0.75h.
They meet in 0.75h. It means that
[tex]S_{1}(0.75) = S_{2}(0.75)[/tex]
With this we find the value o v, and replace in the equation of [tex]S_{2}[/tex] to see how far they are from Vancouver when they meet.
[tex]S_{1}(0.75) = S_{2}(0.75)[/tex]
[tex]0.75(v+2) = 22 - 0.75v[/tex]
[tex]0.75v + 1.50 = 22 - 0.75v[/tex]
[tex]1.50v = 20.50[/tex]
[tex]v = \frac{20.50}{1.50}[/tex]
[tex]v = 13.67[/tex]km/h.
[tex]S_{2}(t) = 22 - vt[/tex]
[tex]S_{2}(0.75) = 22 - 13.67*0.75 = 11.7475[/tex]
They are 11.7475km away from Vancouver when they meet.
The ferries are 10.25 km away from Vancouver when they meet after traveling for 45 minutes, with the ferry from Nanaimo moving at 15.67 km/h and the one from Vancouver at 13.67 km/h.
Distance Between Ferries
Let's denote the speed of the slower ferry as s km/h. Therefore, the speed of the faster ferry leaving Nanaimo will be s + 2 km/h. As they both start at the same time, we can express the distance they travel in terms of their speeds multiplied by the time, which is 45 minutes (or 0.75 hours when converted to hours).
Now, let's set up the equation for the distance each ferry has traveled when they meet: the distance traveled by the slower ferry plus the distance traveled by the faster ferry equals the total distance between Nanaimo and Vancouver, which is 22 km.
Distance of slower ferry: s (0.75) km
Distance of faster ferry: (s + 2)(0.75) km
The sum of both distances is 22 km, so we have:
s (0.75) + (s + 2)(0.75) = 22
Rearranging terms and solving for s gives us:
1.5s + 1.5 = 22
s = (22 - 1.5)/1.5
Calculating the value of s and consequently the distance each ferry traveled before meeting will give us the answer.
After solving, we can determine that the speed of the slower ferry is 13.67 km/h and the faster ferry is 15.67 km/h. To find the distance from Vancouver to the meeting point, we only need to calculate the distance traveled by the slower ferry:
Distance from Vancouver = (13.67 km/h) (0.75 h) = 10.25 km
Therefore, the ferries are 10.25 km away from Vancouver when they meet.
(x-5)^2=81, solve for x
Answer:
x ∈ {-4, 14}
Step-by-step explanation:
Take the square root and add 5.
(x -5) = ±√81 = ±9
x = 5 ± 9
x ∈ {-4, 14} . . . . . . x may be either of -4 or 14
The point P(1, 1/6) lies on the curve y = x/(5 + x). If Q is the point (x, x/(5 + x)), use a scientific calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
Answer:
[tex]m(x)=\frac{5}{6(5+x)}[/tex]
Step-by-step explanation:
Slope of the secant line PQ:
P : (1, 1/6)
Q : (x, x/(5 + x))
[tex]m(x)=\frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}=\frac{x/(5 + x)-1/6}{x-1}=\frac{5(x-1)}{6(5+x)(x-1)}[/tex]
The slope of the secant line PQ is 0.
Explanation:To find the slope of the secant line PQ, we need to determine the coordinates of Q and P. Given that Q has the coordinates (x, x/(5+x)), we can substitute x=1 into the equation to find Q. So, Q is (1, 1/6).
The slope of a line passing through two points can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates of P and Q into the formula:
m = (1/6 - 1/(5+1)) / (1 - 1) = (1/6 - 1/6)/(0) = 0
Therefore, the slope of the secant line PQ is 0.
Assume that the readings on the thermometers are normally distributed with a mean of 0 C and a standard deviation of 1 C. A thermometer is randomly selected and tested. If 1.7% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others.
Answer: -2.12°C
Step-by-step explanation:
Let x denotes the reading of the thermometers .
We assume that the readings on the thermometers are normally distributed.
Let a be the reading that separates the rejected thermometers from the others.
Given: Population mean : [tex]\mu=0[/tex]
Standard deviation: [tex]\sigma= 1[/tex]
Also, [tex]P(x<a)=0.017[/tex]
By using the z-table , the z-value corresponds to the p-value (one -tailed)0.017 is [tex]\pm2.12[/tex].
Now, [tex]z=\dfrac{a-\mu}{\sigma}[/tex]
i.e. [tex]\pm2.12=\dfrac{a-0}{1}[/tex]
i.e. [tex]\pm2.12=a[/tex]
For left tailed , [tex]a=-2.12[/tex]
It means the reading that separates the rejected thermometers from the others = -2.12°C.
calculate the perimeter of a rectangle with a length of 17.5 cm and a width of 40 mm in cm
Answer: 43 cm
Step-by-step explanation:
The perimeter of rectangle is given by :-
[tex]P=2(l+w)[/tex], where l is length and w is width of the rectangle.
Given : The length of rectangle is 17.5 cm and the width is 40 mm in cm.
Since , 1 cm = 10 mm
Then, 40 mm= [tex]\dfrac{40}{10}\ cm=4\ cm[/tex]
Then, the perimeter of rectangle will be :-
[tex]P=2(17.5+4)\\\\\Rightarrow\ P=2(21.5)\\\\\Rightarrow\ P=43\ cm[/tex]
Hence, the perimeter of rectangle = 43 cm
The purpose of this assignment is to make sures linear equations. This skill is required when Chapter 5 is covered. Problem Statement Given is a linear Equation 1 in the format of y mx+b. Equation 1, y = 46.888 +0.78X Plot Equation 1 on a graph paper. Label all axes and markers on the graph. 1. Save your graph in a pdf file zero point is issued for non pdf file format 2. Click on the assignment link. W01 A3a. below to upload your pdf file via attachment by the due date. 3. If you do not see the assignment link it is because the assignment is past due. Wk01 A3a De as
Answer: The plot will be linear as described by the equation of the problem
Step-by-step explanation:
The first thing to do is to determine values of X and place them in one column. The following column should include all values solved by the equation in terms of y. Once you plot the graph, you should have something as the graph attached. To save it as a pdf, simply scan your graph paper to pdf and follow the instructions for assignment submission.
A telemarketer makes a sale on 25% of his calls. If he makes 300 calls in a night, what is the probability that he will make more than 70 sales but less than 90 sales?
Answer: 0.7258
Step-by-step explanation:
Given : A telemarketer makes a sale on 25% of his calls.
i.e. p=0.25
He makes 300 calls in a night, i.e. n=300
Let x be a random variable that represents the number of calls make in night.
To convert the given binomial distribution to normal distribution we have :-
[tex]\mu=np=300(0.25)=75[/tex]
[tex]\sigma=\sqrt{p(1-p)n}=\sqrt{(0.25)(1-0.25)(300)}\\\\=\sqrt{56.25}=7.5[/tex]
Now, using [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-value corresponds to x= 70 :-
[tex]z=\dfrac{70-75}{7.5}\approx-0.67[/tex]
The z-value corresponds to x= 90 :-
[tex]z=\dfrac{90-75}{7.5}\approx2[/tex]
By using the standard normal distribution table for z, the probability that he will make more than 70 sales but less than 90 sales:-
[tex]P(-0.67<z<2)=P(z<2)-P(z<-0.67)\\\\=P(z<2)-(1-P(z<0.67))\\\\=0.9772-(1-0.7486)\\\\=0.9772-0.2514=0.7258[/tex]
Hence, the probability that he will make more than 70 sales but less than 90 sales= 0.7258
What is the area of the region bounded between the curves y=6x^2 and y=4x?
Answer:
The area of the region between the curves y=6x^2 and y=4x is 8/27
Step-by-step explanation:
Use the diagram to visualize the problem, the area colored of blue is the one that needs to be found, let's do it in 3 parts:
Part 1: Find the intersection points of the curves
To do this we put both equations in one and solve it for x:
[tex]6x^2=4x[/tex]
[tex]6x^2-4x=0\\2x(3x-2)=0[/tex]
This equation has 2 possible solutions:
x=0 and x=2/3, so the interval for integration is 0 <= x <= 2/3
Part 2: Find the area below each curve
[tex]A_{blue}=\int\limits^0_{2/3} {6x^2} \, dx \\A_{blue}=2x^3[/tex], evaluate in 0 and 2/3
[tex]A_{blue}=\frac{16}{27}[/tex]
[tex]A_{red}=\int\limits^0_{2/3} {4x} \, dx \\A_{red}=2x^2[/tex], evaluate in 0 and 2/3
[tex]A_{red}=\frac{8}{9}[/tex]
Part 3: Substract the area of the blue curve (y=6x^2) to the area of the red curve (y=4x)
[tex]Area=\frac{8}{9}-\frac{16}{27}\\Area=\frac{8}{27}[/tex]
Researchers measured the data speeds for a particular smartphone carrier at 50 airports. The highest speed measured was 75.6 Mbps. The complete list of 50 data speeds has a mean of x overbar equals 15.62 Mbps and a standard deviation of s equal s 20.03 Mbps.
a. What is the difference between carrier's highest data speed and the mean of all 50 data speeds?
b. How many standard deviations is that [the difference found in part (a)]?
c. Convert the carrier's highest data speed to a z score.
d. If we consider data speeds that convert to z scores between minus 2 and 2 to be neither significantly low nor significantly high, is the carrier's highest data speed significant?
Answer:
a) 59.98
b) 2.99
c) 2.99
d) Significantly High
Step-by-step explanation:
Part a)
Highest speed measured = x = 75.6 Mbps
Average/Mean speed = [tex]\overline{x}[/tex] = 15.62 Mbps
Standard Deviation = s = 20.03 Mbps
We need to find the difference between carrier's highest data speed and the mean of all 50 data speeds i.e. x - [tex]\overline{x}[/tex]
x - [tex]\overline{x}[/tex] = 75.6 - 15.62 = 59.98 Mbps
Thus, the difference between carrier's highest data speed and the mean of all 50 data speeds is 59.98 Mbps
Part b)
In order to find how many standard deviations away is the difference found in previous part, we divide the difference by the value of standard deviation i.e.
[tex]\frac{59.98}{20.03}=2.99[/tex]
This means, the difference is 2.99 standard deviations or in other words we can say, the Carrier's highest data speed is 2.99 standard deviations above the mean data speed.
Part c)
A z score tells us that how many standard deviations away is a value from the mean. We calculated the same in the previous part. Performing the same calculation in one step:
The formula for the z score is:
[tex]z=\frac{x-\overline{x}}{s}[/tex]
Using the given values, we get:
[tex]z=\frac{75.6-15.62}{20.03}=2.99[/tex]
Thus, the Carriers highest data is equivalent to a z score of 2.99
Part d)
The range of z scores which are neither significantly low nor significantly high is -2 to + 2. The z scores outside this range will be significant.
Since, the z score for carrier's highest data speed is 2.99 which is well outside the given range, i.e. greater than 2, we can conclude that the carrier's highest data speed is significantly higher.
Final answer:
The highest data speed recorded is 59.98 Mbps above the mean, and this difference equals approximately 2.996 standard deviations. This results in a z-score of 2.996, indicating that the carrier's highest data speed is significantly high.
Explanation:
a. The difference between the carrier's highest data speed and the mean of all 50 data speeds is found by subtracting the mean speed from the highest speed. This is calculated as 75.6 Mbps - 15.62 Mbps = 59.98 Mbps.
b. To find out how many standard deviations this difference is, we divide the difference by the standard deviation of the data speeds: 59.98 Mbps / 20.03 Mbps = 2.996 standard deviations.
c. The z-score for the carrier's highest data speed is calculated by subtracting the mean from the data speed and then dividing by the standard deviation: (75.6 Mbps - 15.62 Mbps) / 20.03 Mbps = 2.996.
d. Since the z-score of 2.996 is greater than 2, it suggests that the carrier's highest data speed is significantly higher than what is considered neither significantly low nor high. In other words, the carrier's highest data speed is significant.
Solve the equation |x-2|-3=0 first by finding the zeros of
y=|x-2|-3 and then algebraically.
Answer:
The solution of given equation are -1 and 5.
Step-by-step explanation:
The given equation is
[tex]|x-2|-3=0[/tex]
We need to solve the above equation by finding the zeros of
[tex]y=|x-2|-3[/tex]
The vertex form of an absolute function is
[tex]y=a|x-h|+k[/tex]
where, a is constant and (h,k) is vertex.
Here, h=2, k=-3. So vertex of the function is (2,-3).
The table of values is
x y
0 -1
2 -3
4 -1
Plot these points on a coordinate plane and draw a V-shaped curve with vertex at (2,-3).
From the given graph it is clear that the graph intersect x-axis at -1 and 5. So, zeroes of the function y=|x-2|-3 are -1 and 5.
Therefore the solution of given equation are -1 and 5.
Now solve the given equation algebraically.
[tex]|x-2|-3=0[/tex]
Add 3 on both sides.
[tex]|x-2|=3[/tex]
[tex]x-2=\pm 3[/tex]
Add 2 on both sides.
[tex]x=\pm 3+2[/tex]
[tex]x=3+2[/tex] and [tex]x=-3+2[/tex]
[tex]x=5[/tex] and [tex]x=-1[/tex]
Therefore the solution of given equation are -1 and 5.
One group of scientists (Group A) has measured a time for a particular chemical reaction to be completed to be 7.34 ± 0.05 s. A second group of scientists (Group B) complete a similar experiment and measure the time to be 7.38 ± 0.03s. A third group (Group C) measured the time to be 7.46 ±0.06s. Do results of Group A and B agree with each other within the experimental uncertainty? Do results of Group A and C agree with each other within the experimental uncertainty?
Answer:
1. Group A and B agree with each other.
2. Group A and C do not agree with each other.
Step-by-step explanation:
When we are analizing this problem, we will see what are the ranges of this measured times. Since we are taking into account the error we can see that :
Group A varies from 7.34-0.05 to 7.34+0.05. So the limits are (7.29 ;7.39)Group B varies from 7.38-0.03 to 7.38+0.03. So the limits are (7.35; 7.41)Group C varies from 7.46-0.06 to 7.46+0.06. So the limits are (7.40; 7.52)Question 1 is about the overlapping response in Group A and Group B. And yes, we have an overlap between 7.35 to 7.39. Among this times both group A and B are in agree with each other within the experimental uncertainty.
Question 2 is now referring to Group A and Group C. And no, there isn't any common time where both groups agree with each other.
Describe how an infeasible solution occurs
Answer:
Described
Step-by-step explanation:
A solution becomes infeasible when no solution exit and which satisfies all the constraints. We will consider two basic types of infeasibility. The 1st we will call continuous infeasibility and the second one is discrete or integer infeasibility. Continuous infeasibility is the one where a non–MIP problem is infeasible. In this case the feasible region defined by the intersecting constraints is empty. Discrete or integer infeasibility is the one where a MIP problem has a feasible relaxation (note that a relaxation of a MIP is the problem we get when we drop the discreteness required on the variables) but the feasible region of the relaxation contains no solution that satisfies the discreteness requirement.
Final answer:
An infeasible solution occurs when constraints of a problem conflict with each other, making it impossible to find a solution that satisfies all conditions. This is common in linear programming and can happen due to poor design decisions or unreasonable premises. Brainstorming alternative solutions is essential for finding feasible approaches.
Explanation:
An infeasible solution occurs in problem-solving when the constraints of a problem are such that no possible solution satisfies all the conditions. In mathematics and particularly in linear programming, this can happen when the set of constraints creates a situation where no intersection point exists between all the constraint equations - essentially, the requirements for a solution are mutually exclusive and cannot be met simultaneously.
For example, a situation may arise where a design problem contains a premise that produces an unreasonable result, highlighting that some design choices may not align with the physical reality, and thus are infeasible. An extremely large Coulomb force in a physics problem, due to the assumption of an excessively large separated charge, is an instance where the result is obviously incorrect and infeasible based on known physical principles.
When approaching such problems, a common strategy is to brainstorm alternative solutions and explore a wide variety of ideas that could potentially solve the problem within realistic constraints. Some ideas may be discarded as infeasible early on if they conflict with the basic laws of nature or if they stand little chance of practical application. This process is especially useful in engineering and design fields, where practicality and adherence to real-world principles are crucial.
15.) A market
research worker interviewed a random sample of 18people about their
use of a certain product. The results, in termsof Yes (Y) or No (N)
are as follows:Y-N-N-Y-Y-Y-N-Y-N-Y-Y-Y-N-Y-N-Y-Y-N. Estimate the
populationproportion of users of the product.
Answer: [tex]p=0.610[/tex]
Step-by-step explanation:
Given : A market research worker interviewed a random sample of 18 people about their use of a certain product.
The results, in terms of Yes (Y) or No (N) are as follows:
Y-N-N-Y-Y-Y-N-Y-N-Y-Y-Y-N-Y-N-Y-Y-N.
The number of people said "Yes" for the product= 11
Then, the sample proportion for the users of the product =[tex]\hat{p}=\dfrac{11}{18}0.611111111111\approx0.61[/tex]
We know that the sample proportion is the best estimate for the population proportion.
Thus the point estimate for population proportion : [tex]p=\hat{p}=0.610[/tex]
The propositional variables, p, q, and s have the following truth assignments: p = T, q = T, s = F. Give the truth value for each proposition. (a) p ∧ ¬(q ∨ s) (b) ¬(q ∧ p ∧ ¬s)
Answer: a) FALSE b) FALSE
Step-by-step explanation:
a) For the given proposition p ∧ ¬(q ∨ s) you can solve first (q v s)
q v s is true if either q is true or s is true or both. It is only false if both q and s are false. So, the proposition (q v s) is true because q is true.
Now you can solve the negation: ¬(q ∨ s)
As we know, (q v s) is true then its negation ¬(q ∨ s) is false.
p ∧ ¬(q ∨ s) should be true when both p and ¬(q ∨ s) are true, and false otherwise. So, the proposition is false because p is true and ¬(q ∨ s) is false.
b) For the given proposition ¬(q ∧ p ∧ ¬s)
You can rewrite the expression as: ¬[q ∧ (p ∧ ¬s) ] to solve first each part of the propositions in parenthesis.
The negation of s: ¬s is true because s is false
Now, you can solve (p ∧ ¬s) which is true because both p and ¬s are true.
To continue, you have to solve (q ∧ p ∧ ¬s) which is true because both q and (p ∧ ¬s) are true.
To finish, the negation: ¬(q ∧ p ∧ ¬s) is false.
A cell phone company offers two different plans. Plan A costs $99 per month for unlimited talk and text. Plan B costs $0.20 per minute plus $0.10 per text message sent. You need to purchase a plan for your teenage sister. Your sister currently uses 1,750 minutes and sends 1,700 texts each month. What is your sister’s total cost under each of the two plans?
Which correlation coefficient corresponds to the best-fit line that most closely models it’s set of data. A. -0.87, B. -0.15. C. 0.13. D. 0.84
Answer:
A. -0.87
Step-by-step explanation:
The correlation coefficient is a measure of the strength and nature of the linear relationship between two variables, one dependent and one independent. This coefficient takes values between -1 and 1, indicating with its sign whether the relationship is direct or inverse between the variables involved and with its absolute value indicates the strength of the linear relationship between them. A coefficient with absolute value close to 1 indicates great strength and better fit.
Conclusion: The best coefficient, of the propuetso in the problem, is that of -0.87, which indicates a strong relationship between the variables, a good fit and an inverse relationship between them.
Use a direct proof to show that the product of two odd integers is odd.
Step-by-step explanation:
A direct proof is a method that takes an statement p, which we assume to be true, and use it to show directly that another statement q is true. So this method has the following steps:
Assume the statement p is trueUse what we know about p and other facts as necessary to deduce that another statement q is true, that is show p ⇒ q is true.Fact that we need to use:
Every odd integer can be written in the form 2m + 1 for some unique other integer m
Let p be the statement a and b be odd integers and q be the statement that the product of a and b is odd.
Proposition if a and b are odd, then the product of a and b is odd
Proof: Assume that a and b are odd integers, the by definition a = 2m + 1 and b = 2n + 1 for some integers m and n. we will now use this to show that the product of a and b is odd.
[tex]a\cdot b= (2m+1) \cdot (2n+1)\\a\cdot b = 2m\cdot 2n+2m+2n+1\\a\cdot b =4mn+2m+2n+1\\a\cdot b = 2(2mn+2m+2n) +1\\\:If \:k=2mn+2m+2n\\a\cdot b = 2k+1[/tex]
Hence we have shown that the product of a and b is odd since 2k + 1 is and odd integer. Therefore we have shown that p ⇒ q and so we have completed our proof.
Answer:
Step-by-step explanation:
The proof by the direct method that the product of two odd numbers is odd integer number, is the following:
Let [tex]z_1[/tex] and [tex]z_2[/tex] be two odd integers, then [tex]z_1 = 2a+1[/tex] and [tex]z_2 = 2b +1[/tex], for some integers a and b.
[tex]z_1z_2 = (2a + 1) (2b + 1)\\\\z_1z_2 = 4ab + 2a + 2b + 1\\\\z_1z_2 = 2 (2ab + a + b) +1\\\\z_1z_2 = 2n + 1[/tex]
where [tex]n = 2ab + a + b[/tex], which guarantees that [tex]n[/tex] is an integer number. In this way, [tex]z_1z_2[/tex] is an odd integer.
Tim has just inherited £1000 and he decides not to invest it but rather to spend 10% of the remaining money each month. Each month Tim's saving thus reduce by a factor of 0.9 Calculate, to the nearest integer, the amount remaining (in E) after 20 months. You may find it useful to first devise a formula for the amount remaining after n months. Enter your answer, to the nearest integer and without units, in the box below. Answer:
Answer:
£ 121.57
Step-by-step explanation:
As given in question,
Total amount inherited by Tim = £1000
He spends each month = 10 % of remaining money
savings reduced each month = 0.9 of remaining amount
So, the amount of money after one month = 0.9 x £1000
amount of money remained after 2 months = 0.9 x 0.9 x £1000
amount of money remained after 3 months = 0.9 x 0.9 x 0.9 x £1000
Hence, the amount of money remained after the n months can be given by,
[tex]E\ =\ 0.9^n\times 1000[/tex]
Hence, amount of money remained after 20 months can be given by,
[tex]E_{20}\ =\ 0.9^{20}\times1000[/tex]
= £ 121.57
So, the amount of money remained after 20 months will be £ 121.57.
A sales representative for a Children's Fashion store earns a salary of $1100.00 per month plus a commission based on the total sales. During the month of January, this sales representative's total sales were $8300.00 and was paid $2130.00. What is this sales representative's commission rate?
a.
14.30963855%
b.
11.10963855%
c.
13.60963855%
d.
12.40963855%
e.
14.60963855%
f.
None of the above.
Answer:
Option d. 12.40963855% is the answer.
Step-by-step explanation:
Monthly salary of a sales representative = $1,100
He was paid for the month = $2,300
His total sales were = $8,300
His commission for the month of sales = Total payment - monthly salary
= 2,300 - 1,100
= $1,030
He got $1,030 as commission on the sales of $8,300.
The percentage of the commission = [tex]\frac{1030}{8300}\times 100[/tex]
= 12.40963855421687%
Option d. 12.40963855% is the answer.
A normal curve with mean = 25 has an area of .3531 between 25 and 34. What is the standard deviation for this normal curve?
Answer:
The desviation is 8 4/7 or 8.571
Step-by-step explanation:
The conversion for any variable X to a standard z is
[tex]Z=\frac{X-\[Mu]}\\{\[Sigma]}[/tex]
where mu is the mean and sigma de desviation
You can find the value of Z whith the tables of the accumulated probability function. The accumulated probability for the mean is 0.5 .Remenber that the accumlated probability function represent the area at the left of an abscissa. Then
0.5+0.3531=0.8531
Acording to the accumulated probability function table, a Z=1.05 has an area of 0.8531 at its left.
Now it is only solving the equation
[tex]\[Sigma]=\frac{X-\[Mu]}{Z}[/tex]
σ=[tex]\frac{34-25}{1.05}[/tex]
σ=8.571
Solve the differential equation x^2 y"-xy' +y 0
Answer:
[tex]y(x)\ =\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
Step-by-step explanation:
Given differential equation is
[tex]x^2y"-xy'+y=0[/tex] (1)
Let's assume that
[tex]x=e^t[/tex]
[tex]=>\ t\ =\ logx[/tex]
then,
[tex]\dfrac{dx}{dt}=e^t[/tex]
[tex]and\ \dfrac{d^2x}{dt^2}=e^t[/tex]
We can write,
[tex]\dfrac{dy}{dx}=\dfrac{dy}{dt}.\dfrac{dt}{dx}[/tex]
[tex]=e^{-t}\dfrac{dy}{dt}[/tex]
Similarly,
[tex]\dfrac{d^2y}{dt^2}=\dfrac{d^2y}{dt^2}.\dfrac{dt^2}{dx^2}[/tex]
[tex]=e^{-2t}.\dfrac{d^2y}{dt^2}[/tex]
Putting these values in equation (1), we will get
[tex]e^{2t}.e^{-2t}.\dfrac{d^y}{dt^2}-e^t.e^{-t}\dfrac{dy}{dt}+y=0[/tex]
[tex]=>\dfrac{d^2y}{dt^2}-\dfrac{dy}{dt}+y=0[/tex]
So, the characteristics equation can be given as
[tex]D^2-D+1=0[/tex]
[tex]=>D\ =\ \dfrac{1+\sqrt{1-4}}{2}\ or\ \dfrac{1-1\sqrt{1-4}}{2}[/tex]
[tex]=>D=\ \dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\ or\ \dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}[/tex]
Hence, the general solution of the equation can be give by
[tex]y(t)\ =\ e^{\dfrac{t}{2}}[C_1cos\dfrac{\sqrt{3}}{2}t+C_2sin\dfrac{\sqrt{3}}{2}t][/tex]
Now, by putting the value of t in above solution, we will have
[tex]y(x)\ =\ e^{\dfrac{1}{2}logx}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
Hence, the solution of above given differential equation can be given by
[tex]y(x)=\ \sqrt{x}[C_1cos\dfrac{\sqrt{3}}{2}logx+C_2sin\dfrac{\sqrt{3}}{2}logx][/tex]
Show that any integer n > 12 can be written as a sum 4r + 5s for some nonnegative integers r, s. (This problem is sometimes called a postage stamp problem. It says that any postage greater than 11 cents can be formed using 4 cent and 5 cent stamps.)
Answer:
Use induction for the prove
Step-by-step explanation:
Mathemathical induction is an useful method to prove things over natural numbers, you check for the first case, supose for the n and prove using your hypothesis for n+1
there says any integer bigger than 12 can be written as 4r+5s
so first number n can be is 13.
we can check n=13 = 4*2+5*1 r=2 and s=1 give 13.
Now we suppose n can be written as 4r+5s
and we can check if n+1=4r'+5s' with r' and s' integers.
we replace n as 4r+5s because that is our hypotesis
n+1=4r+5s+1
if we write that 1 as 5-4
4r+5s+1
4r+5s+5-4
then we can write
4(r-1)+5(s+1) , we got n+1= 4 (r-1) +5(s+1) where r-1 and s+1 are non negative integers. because r and s were no negative integers ( if r is not 0)
what if r=0?
if r is 0 , n is a multiple of 5 and n+1 can be written as 5s+1
first multiple of 5 we can write is 15 since n is bigger than 12 , then smaller s is 3.
for any n+1 we can write
n+1=5s+1=5 (s-3) +3*5+1=5(s-3)+4*4, s-3 is 0 or bigger.
(check 3*5+1 is 16, the same as 4*4)
Final answer:
Any integer greater than 12 can be expressed as the sum 4r + 5s with nonnegative integers r and s by providing concrete examples for n values from 13 to 17 and then proving for any n > 17 using number theory.
Explanation:
To show that any integer n greater than 12 can be written as the sum 4r + 5s for some nonnegative integers r and s, we can provide examples and create a general proof. We know that for n = 13 through n = 17, specific values of r and s can be found that satisfy the equation:
n = 13 = 4(1) + 5(1)
n = 14 = 4(3) + 5(0)
n = 15 = 4(0) + 5(3)
n = 16 = 4(4) + 5(0)
n = 17 = 4(1) + 5(3)
For n > 17, we can write n = 17 + k, where k is a nonnegative integer. Since every nonnegative integer can be expressed as a multiple of 4 plus an additional 0, 1, 2, or 3 (as k = 4q + r, with 0 ≤ r < 4), we can substitute into our equation for n to n = 17 + 4q + r. The key is to notice that for any additional 4 we add to the sum, we can simply increase r (our existing count of 4-cent stamps), ensuring that the equation 4r + 5s will always hold.
This proof shows that we can always add enough 4-cent stamps (or subtract them and add a 5-cent stamp) to reach any postage amount above 12 cents. This is a classic example of a problem that uses the concept of number theory and diophantine equations.
The Big Wave Surf Shop sells 8 ounce containers of sunscreen for $8 each. The total daily cost associated with the sale of this item yesterday was $200. What is the shop’s profit if it sold 150 units of sunscreen yesterday?
Answer:
The shop’s profit if it sold 150 units of sunscreen yesterday is $1000
Step-by-step explanation:
The total daily cost associated with the sale of this item yesterday was $200.
Cost of 1 unit = $8
So, Cost of 150 units = [tex]8 \times 150[/tex]
= [tex]1200[/tex]
So, revenue = $1200
Cost = $200
So Profit = Revenue - Cost
Profit = $1200-$200
Profit = $1000
Hence the shop’s profit if it sold 150 units of sunscreen yesterday is $1000
Determine which of the following sets of three points constitute the vertices of a right triangle: (a) 3 + 5i,2 +2i,5i; (b)2i,3 + 5i,4 + i; (c)6 +4i,7 + 5i, 8 +4i
Answer:
Option (c) is correct
Step-by-step explanation:
Case (a)
A = 3 + 5i = (3, 5)
B = 2 + 2i = (2, 2)
C = 5i = (0, 5)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(2-3)^{2}+(2-5)^{2}}=\sqrt{10}[/tex]
[tex]BC = \sqrt{(0-2)^{2}+(5-2)^{2}}=\sqrt{13}[/tex]
[tex]CA = \sqrt{(0-3)^{2}+(5-5)^{2}}=\sqrt{9}[/tex]
For the triangle to be right angles triangle
[tex]BC^{2}=AB^{2}+CA^{2}[/tex]
Here, it is not valid, so these are not the points of a right angled triangle.
Case (b)
A = 2i = (0, 2)
B = 3 + 5i = (3, 5)
C = 4 + i = (4, 1)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(3-0)^{2}+(5-2)^{2}}=\sqrt{18}[/tex]
[tex]BC = \sqrt{(4-3)^{2}+(1-5)^{2}}=\sqrt{17}[/tex]
[tex]CA = \sqrt{(4-0)^{2}+(1-2)^{2}}=\sqrt{17}[/tex]
For the triangle to be right angles triangle
[tex]AB^{2}=BC^{2}+CA^{2}[/tex]
Here, it is not valid, so these are not the points of a right angled triangle.
Case (c)
A = 6 + 4i = (6, 4)
B = 7 + 5i = (7, 5)
C = 8 + 4i = (8, 4)
Use the distance formula to find the distance between two points
[tex]AB = \sqrt{(7-6)^{2}+(5-4)^{2}}=\sqrt{2}[/tex]
[tex]BC = \sqrt{(8-7)^{2}+(4-5)^{2}}=\sqrt{2}[/tex]
[tex]CA = \sqrt{(8-6)^{2}+(4-4)^{2}}=\sqrt{4}[/tex]
For the triangle to be right angles triangle
[tex]CA^{2}=BC^{2}+AB^{2}[/tex]
Here, it is valid, so these are the points of a right angled triangle.
An electricity company charges its customers a fixed base charge of $6 a month, plus 10 cents per kilowatt-hour (kWh) for the first 500 kWh, 11 cents per kWh for the next 500 kWh, and 15 cents for all additional kWh. Express the monthly cost E as a function of the amount x of electricity used.
Answer:
[tex]E(k)= \left[ \begin{array}{c} {6+0.10k\,\,if\,\,0\leq k \leq500} & 6+500\times0.10+0.11(k-500)\,\,if\,\,500<k\leq1000& 6+500\times0.10+500\times0.11+0.15(k-1000)\,\,if\,\,k>1000\end{array}[/tex]
Step-by-step explanation:
The cost function has 3 branches,
Consumptions less than 500Consumptions less than 1000consumptions above a 1000So the first branch the consumer pays 6 dollars plus .10cents for any additional kWh (k)
In the second, they pay the same as the first up to 500kWh, and after that they pay 0.11 for the additional kWh above 500: (k-500) but bellow 1000
In the third branch, for consumptions above 1000, they pay the fix amount, plus .10 for the first 500 ([tex]500\times0.10[/tex]) , .11 for the additional 500 ([tex]500\times0.11[/tex]) , and finally 0.15 for consumptions above 1000: [tex]0.15(k-1000)[/tex]
Final answer:
The monthly cost of electricity, E, as a function of the electricity used, x, is defined using a piecewise function with different rates for usage tiers, including a fixed base charge and variable rates for different consumption brackets.
Explanation:
To express the monthly cost E as a function of the amount x of electricity used by the customer, we consider the following rates:
A fixed base charge of $6 per month.10 cents per kilowatt-hour (kWh) for the first 500 kWh.11 cents per kWh for the next 500 kWh (501 to 1000 kWh).15 cents per kWh for all additional kWh (above 1000 kWh).Therefore, the function is a piecewise function defined as:
E(x) =
6 + 0.10x if 0 ≤ x ≤ 5006 + (0.10 × 500) + 0.11(x - 500) if 500 < x ≤ 10006 + (0.10 × 500) + (0.11 × 500) + 0.15(x - 1000) if x > 1000This function helps calculate the monthly electricity bill based on the usage levels. Remember that the first bracket is for the initial 500 kWh, the second bracket is for 501 to 1000 kWh, and the last bracket applies to usage above 1000 kWh.