Answer:
a, b and c= (-∞,∞) d = (-∞,3) U (-3,3) U (3,∞)
Step-by-step explanation:
Hi there!
1) Firstly, let's recap the sum of functions rule:
f(x) + g(x) = (f+g)(x)
Applying it to those functions, we have:
f(x)=5-x +g(x)=[tex]x^{2} -9[/tex] = [(5-x) +([tex]x^{2}[/tex]-9)](x)
(f+g)(x)=[tex]5-x+x^{2} -9[/tex]
2) To State the Domain is to state the set which is valid the quantities of x, of a function. In this case,
a) (f+g)(x)= 5-x+[tex]x^{2}[/tex]+9
Simplifiying
(f+g)(x) =[tex]x^{2} -x+4=0[/tex]
Since there are no restrictions neihter discontinuities, this function has a Domain which can expressed this way:
X may assume infinite quantities, negatives or positives one in the Real set.
(-∞< x <∞+) or simply (-∞,∞)
Or simply put, x ∈ R. Remember, ∞ is not a number, it's a notation meaning infinite values. That's why it's not a closed interval.
Check the graph below.
b) (f-g)(x) =(5-x) -(x²-9)
(f-g)(x)= 5-x-x²+9
Domain of (f-g)(x) =x²-x+14
Similarly to a) this function (f-g)(x) has not discontinuity, nor restrictions on its Domains.
Since there are no restrictions either discontinuities, this function has a Domain which can be expressed this way:
X may assume infinite quantities, negatives or positives one in the Real set.
(-∞< x <∞ +) then finally, the answer: (-∞,∞)
c) (f*g)(x)=(5-x)(x²-9)
(f*g)(x)=5x²-45-x³+9x
Again, this function has no discontinuities, nor restrictions in its Domain as you can check it on its graph.
Then, the Domain of (f*g)(x)=(5-x)(x²-9) is also (-∞,∞)
d) (f/g)(x) =(5-x)/(x²-9)
Highlighting the denominator, we can calculate the Domain.
We can see a restriction here. There is no denominator zero, defined for the Set of R.
Then, let's calculate
[tex]x^{2} -9>0\\ \sqrt{x^2} >\sqrt{9}\\x>3[/tex] and x < -3
In the Numerator, no restrictions.
So the Domain will be the union between the Numerator's Domain and the Denominator's Domain with Restrictions.
Check the graph below.
Finally
D = (-∞,3) U (-3,3) U (3,∞)
Final answer:
The sum, difference, product, and quotient of two functions f(x) = 5 - x and g(x) = x^2 - 9 can be found by adding, subtracting, multiplying, and dividing their corresponding terms. The domain of each resulting function can be determined by considering the domain of the original functions and any restrictions imposed by the operations involved.
Explanation:
(a) f + g
To find the sum of two functions, we simply add the corresponding terms. So, f + g = (5 - x) + (x^2 - 9).
Simplifying, we get f + g = x^2 - x - 4.
The domain of the function f + g is the intersection of the domains of f and g. Since both f and g are defined for all real numbers, the domain of f + g is also all real numbers, which can be represented as (-∞, ∞) in interval notation.
(b) f - g
To find the difference of two functions, we subtract the corresponding terms. So, f - g = (5 - x) - (x^2 - 9).
Simplifying, we get f - g = -x^2 + x + 14.
Similar to part (a), the domain of the function f - g is all real numbers, which can be represented as (-∞, ∞) in interval notation.
(c) fg
To find the product of two functions, we multiply the corresponding terms. So, fg = (5 - x)(x^2 - 9).
Simplifying, we get fg = -x^3 + 9x - 5x^2 + 45.
Similar to parts (a) and (b), the domain of the function fg is all real numbers, which can be represented as (-∞, ∞) in interval notation.
(d) f/g
To find the quotient of two functions, we divide the corresponding terms. So, f/g = (5 - x)/(x^2 - 9).
However, we need to consider the values of x that make the denominator zero, since division by zero is undefined. In this case, the denominator x^2 - 9 is equal to zero when x = ±3.
Therefore, the domain of the function f/g is all real numbers except x = ±3. In interval notation, this can be represented as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).
Find the average of the ff: 1. 10, 12, 16, 17,20 2. 0, 3,6,9
Answer:
1) 15
2) 4.5
Step-by-step explanation:
1 ) 10, 12, 16, 17,20
Average = [tex]\frac{\text{Sum of all observations}}{\text{Total no. of observations}}[/tex]
Average = [tex]\frac{10+12+16+17+20}{5}[/tex]
Average = [tex]15[/tex]
2) 0, 3,6,9
Average = [tex]\frac{\text{Sum of all observations}}{\text{Total no. of observations}}[/tex]
Average = [tex]\frac{0+3+6+9}{4}[/tex]
Average = [tex]4.5[/tex]
Write a complete mathematical definition of (a,b), the greatest common divisor of a,b in \mathbb{Z}
Answer:
The greatest common divisor of two integers a and b (not both 0) is the largest integer that divides both a and b.
Step-by-step explanation:
Think for example of the numbers a=5, and b= -10. The greatest common divisor of 5 and -10, is the largest integer that divides both 5 and -10. We can find it by inspection (although there are more advanced methods to find it). We can list all integers that divide both 5 and -10.
-5 divides 5, and it also divides -10
-1 divides 5, and it also divides -10
1 divides 5, and it also divides -10
5 divides 5, and it also dividies -10
The LARGEST of them all is then 5, so 5 is the greatest common divisor of 5 and -10. The usual way to write it is
[tex]gcd(5,-10)=(5,-10)=5[/tex]
The greatest common divisor (GCD) of two integers a and b is the largest integer that divides both a and b without leaving a remainder. The GCD, denoted as (a, b), can be calculated using the Euclidean algorithm.
The greatest common divisor (GCD) of two non-zero integers a and b, denoted as (a, b), is the largest positive integer that divides both a and b without leaving a remainder. A formal definition could be formulated as follows: The greatest common divisor of a and b is a number d such that:
d is a divisor of a (i.e., a mod d = 0).
d is a divisor of b (i.e., b mod d = 0).
For any other integer e that divides both a and b, e \\leq d.
Moreover, the GCD can be calculated using algorithms such as the Euclidean algorithm, which is based on the principle that the gcd of a and b is the same as the gcd of b and a mod b, assuming a > b and b \\neq 0.
11101010100110.0101010111 base 2 (convert this number to base 4)
Answer:
3222212.11113
Step-by-step explanation:
First, you should take care of the fractional separator (the dot) so we split the problem in two parts: one for the integer and other for the fractional part.
Since 4 is a power of 2, we can just take two digits from the orignal number and asign it to its corresponding number in base 4:
[tex]\left[\begin{array}{cc}Binary&Base 4\\00&0\\01&1\\10&2\\11&3\end{array}\right][/tex]
Start with the fractional part from the fractional point to the right:
[tex]\left[\begin{array}{ccccc}01&01&01&01&11\\1&1&1&1&3\end{array}\right][/tex]
Then do the same to the integer part starting from the fractional point to the left.
[tex]\left[\begin{array}{ccccccc}11&10&10&10&10&01&10\\3&2&2&2&2&1&2\end{array}\right][/tex]
By joining them together, we obtain the response.
Plz Help if possible
Write the equation of the line, in standard form, that has a y-intercept of 2 and is parallel to 2x + y = -5. Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
Answer:
2x + y = 2.
Step-by-step explanation:
First find the slope of the required line by writing the line 2x + y = -5 in slope intercept form:
2x + y = -5
y = -2x - 5
- so the slope is -2.
Therefore the required equation is
y = -2x + 2 (where 2 is the y-intercept).
Converting to standard form:
y = -2x + 2
2x + y = 2.
The equation of the line parallel to 2x + y = -5 with a y-intercept of 2 is 2x + y = 2.
Explanation:To find the equation of a line parallel to the given line, we must first realize that parallel lines have the same slope. The given equation is 2x + y = -5, which can be rearranged into y = -2x - 5, showing us that the slope of the given line is -2. Therefore, the slope of the line we want to find is also -2.
With a slope of -2 and a y-intercept of 2 (since the line is said to intersect with the y-axis at y=2), the slope-intercept form of the line is y = -2x + 2. However, the question requires the answer in standard form. The standard form is Ax + By = C, where A, B and C are integers and A > 0. To convert our slope-intercept equation to standard form, we will add 2x to both sides, obtaining the final equation as 2x + y = 2.
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please answer 1 and 2 and if you can explain!!! i need help asap i’ll mark brainliest!!!
Answer:
1) A.
2) No
Step-by-step explanation:
1 is A because the shaded line extends over numbers greater than -8 but less than -1. The reason it is greater than or EQUAL to -1 is because the dot above -1 is shaded in.
2 is No because in order to solve this equation, you plug in the numbers from the coordinates into the inequality. An ordered pair is always structured as (x,y), so in this case x = -1 and y = 4. To solve, the first step is to plug the numbers in, and you end up with 4< 2(-1) +5.
Then, simplify by adding and multiplying as needed. Now you will end up with 4<-2 +5. Simplify again. Finally you end up with 4<3. The reason the answer is NO, not a solution is because the statement '4<3' (four is less than three) is false. if the equation had ended up being 4>3, then it would have been true.
Directions: Use system of linear equations
9x-3y=-1
1/5x+2/5y=-1/3
Answer:
The solution to this set of linear equations is:
[tex]x=-\frac{1}{3}\\y=-\frac{2}{3}[/tex]
Step-by-step explanation:
This is a system of two equations with two unknown variables x and y, let's call them
Equation 1: [tex]9x-3y=-1[/tex]
Equation 2: [tex]\frac{1}{5}x+\frac{2}{5}y=-\frac{1}{3}[/tex]
The first step is to solve Equation 1 for y, this means to leave the y alone on one side of the equal
[tex]y= 3x+\frac{1}{3}[/tex]
Then with this equation, you can find the value of x by replacing y in Equation 2
[tex]\frac{1}{5}x+\frac{2}{5}(3x+\frac{1}{3})=-\frac{1}{3}[/tex]
Then simplify this equation to find x
[tex]\frac{1}{5}x+\frac{6}{5}x+\frac{2}{15}=-\frac{1}{3}[/tex]
[tex]\frac{1}{5}x+\frac{6}{5}x=-\frac{1}{3}-\frac{2}{15}[/tex]
[tex]\frac{7}{5}x=-\frac{5}{15}-\frac{2}{15}[/tex]
[tex]\frac{7}{5}x=-\frac{7}{15}[/tex]
Now you solve for x
[tex]x=-\frac{1}{3}[/tex]
Now you use this value of x to find y
[tex]y=3(-\frac{1}{3})+\frac{1}{3}\\y=-\frac{2}{3}[/tex]
You can check if this answer is correct by replacing the values of x and y into Equation 1 or 2, in this case, let's take Equation 1:
[tex]9(-\frac{1}{3})-3(-\frac{2}{3})=-1\\-3+2=-1\\-1=-1\\[/tex]
Given the following prescription formula, what is the ratio strength (nearest whole number) of methylcellulose in the finished product? As a ratio is typically expressed as 1:some number, put ONLY the number in the space provided and NOT the 1: portion. DO NOT include any units. • Progesterone 3.8 g • Glycerin 7 mL • 2% methylcellulose solution 50 mL • Cherry syrup ad 90 mL Your Answer: Answer
Answer:
147
Step-by-step explanation:
Given:
Progesterone = 3.8 g
Glycerin = 7 mL
2% methylcellulose solution 50 mL
Cherry syrup ad = 90 mL
Now,
The total volume of the solution = 7 + 50 + 90 = 147 mL
Also,
2% methylcellulose solution 50 mL is concluded as:
the volume of methylcellulose in the solution is 2% of the total volume of the solution
thus,
volume of methylcellulose = 0.02 × 50 mL = 1 mL
Therefore,
Ratio strength of methylcellulose in the finished product
=[tex]\frac{\textup{volume of methylcellulose}}{\textup{ total volume of the solution}}[/tex]
or
= [tex]\frac{\etxtup{1}}{\textup{ 147}}[/tex]
Hence, the answer according to the question is 147
Consider the area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant. (a) Set up the definite integral needed to find the area. (b) Make an appropriate substitution
[tex]y(t)=8\sin(bt)[/tex] has a period of [tex]\dfrac{2\pi}b[/tex], which is to say one "arch" of the curve occurs over the interval [tex]0\le t\le\dfrac\pi b[/tex].
a. Then the area under one such arch is
[tex]\displaystyle\int_0^{\pi/b}8\sin(bt)\,\mathrm dt[/tex]
b. Substitute [tex]u=bt[/tex], so that [tex]\dfrac{\mathrm du}b=\mathrm dt[/tex]. When [tex]t=0[/tex], [tex]u=0[/tex]; when [tex]t=\dfrac\pi b[/tex], [tex]u=\pi[/tex].
Then the integral is
[tex]\displaystyle\frac1b\int_0^\pi8\sin u\,\mathrm du[/tex]
The required area is [tex]\int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]
Given that,
The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.
We have to find,
Set up the definite integral needed to find the area.
Make an appropriate substitution.
According to the question,
The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.
The curve y(t) = 8sin(bt) has a period of 2π\b, which is one arch of the curve occur over the interval [tex]0\leq t\leq \frac{\pi }{b}[/tex].The area under one arch is given by,
[tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
The required area is [tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
Appropriate Substitute u= bt ,Then,
[tex]\dfrac{du}{b} = dt \\\\when \ t=0, \ and \ u=0\\\\when\ t = \dfrac{\pi }{b}, u = \pi[/tex]
Then,
The required integral is ,
[tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]
The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex].
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The numbers on two consecutively numbered gym lockers have a sum of
135.
What are the locker numbers?
Answer:
67,68
Step-by-step explanation:
Consecutive numbers are the numbers that follow each other. They may be arranged from the smaller to larger or larger to smaller ones.
Some examples of consecutive numbers are 42,43,45,46,... or 67,68,69,70,...
Now, let x be the smaller gym locker number.
The number consecutive to x will be x+1. Thus the larger gym locker number is x+1.
It is given in the question that the sum of these two locker number is 135.
⇒(x)+(x+1) = 135
⇒ 2x + 1 = 135
⇒ 2x = 134
⇒ x =67
Thus, the two consecutive locker number are x = 67 and x+1 = 68.
Final answer:
The two consecutively numbered gym lockers that add up to 135 are 67 and 68.
Explanation:
The locker numbers are 67 and 68.
To find the locker numbers, we can set up an equation where x represents the smaller locker number. Since the lockers are consecutively numbered, the larger locker number is x+1. The sum of the two locker numbers is x + (x + 1) = 135. By solving this equation, we find that x = 67, making the locker numbers 67 and 68.
A new security system needs to be evaluated in the airport. The probability of a person being a security hazard is 0.02. At the checkpoint, the security system denied a person without security problems 1.5% of the time. Also the security system passed a person with security problems 1% of the time. What is the probability that a random person does not pass through the system and is without any security problems? Report answer to 3 decimal places.
Final answer:
Using the given probabilities, we find that the probability is 0.965, or 96.5%.
Explanation:
To find the probability that a random person does not pass through the system and is without any security problems, we need to calculate the complement of two events: a person being a security hazard and the security system denying a person without security problems.
First, let's calculate the probability of a person being a security hazard:
Probability of a person being a security hazard = 0.02
Next, let's calculate the probability of the security system denying a person without security problems:
Probability of the security system denying a person without security problems = 1.5% = 0.015
To find the probability that a person does not pass through the system and is without any security problems, we can use the formula:
Probability = (1 - probability of being a security hazard) * (1 - probability of the security system denying a person without security problems)
Probability = (1 - 0.02) * (1 - 0.015)
Probability = 0.98 * 0.985
Probability = 0.9653
Therefore, the probability that a random person does not pass through the system and is without any security problems is 0.965, or 96.5% (rounded to 3 decimal places).
Finn changes his mind and, from now on, decides to take the normal route to work everyday. On any given day, the time (in minutes) it takes Finn to get to work is normally distributed with mean u = 35 and variance = 81 .
Find the 33rd percentile of the time it takes Finn to get to work on any given day. Do not include any units in your answer.
Recall from Question 6 that on any given day, the time (in minutes) it takes Finn to get to work is normally distributed with mean = 35 and variance =81. Assume the time it takes him to get to work is independent from day to day.
Over the next 2 days, find the probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.
Answer:
The 33rd percentile of the time it takes Finn to get to work on any given day is 31.04 minutes.
There is a 61.92% probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.
Step-by-step explanation:
This can be solved by the the z-score formula:
On a normaly distributed set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a value X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Each z-score value has an equivalent p-value, that represents the percentile that the value X is:
The problem states that:
Mean = 35, so [tex]\mu = 35[/tex]
Variance = 81. The standard deviation is the square root of the variance, so [tex]\sigma = \sqrt{81} = 9[/tex].
Find the 33rd percentile of the time it takes Finn to get to work on any given day. Do not include any units in your answer.
Looking at the z-score table, [tex]z = -0.44[/tex] has a pvalue of 0.333. So what is the value of X when [tex]z = -0.44[/tex].
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.44 = \frac{X - 35}{9}[/tex]
[tex]X - 35 = -3.96[/tex]
[tex]X = 31.04[/tex]
The 33rd percentile of the time it takes Finn to get to work on any given day is 31.04 minutes.
Over the next 2 days, find the probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.
[tex]P = P_{1} + P_{2}[/tex]
[tex]P_{1}[/tex] is the probability that Finn took more than 40.5 minutes to get to work on the first day. The first step to solve this problem is finding the z-value of [tex]X = 40.5[/tex].
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{40.5 - 35}{9}[/tex]
[tex]Z = 0.61[/tex]
[tex]Z = 0.61[/tex] has a pvalue of 0.7291. This means that the probability that it took LESS than 40.5 minutes for Finn to get to work is 72.91%. The probability that it took more than 40.5 minutes if [tex]P_{1} = 100% - 72.91% = 27.09% = 0.2709[/tex]
[tex]P_{2}[/tex] is the probability that Finn took more than 38.5 minutes to get to work on the second day. Sine the probabilities are independent, we can solve it the same way we did for the first day, we find the z-score of
[tex]X = 38.5[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{38.5 - 35}{9}[/tex]
[tex]Z = 0.39[/tex]
[tex]Z = 0.39[/tex] has a pvalue of 0.6517. This means that the probability that it took LESS than 38.5 minutes for Finn to get to work is 65.17%. The probability that it took more than 38 minutes if [tex]P_{1} = 100% - 65.17% = 34.83% = 0.3483[/tex]
So:
[tex]P = P_{1} + P_{2} = 0.2709 + 0.3483 = 0.6192[/tex]
There is a 61.92% probability that Finn took more than 40.5 minutes to get to work on the first day or more than 38.5 minutes to get to work on the second day.
178 meters in 2 seconds = meters in 10 seconds
Answer:
178 meters in 2 seconds = 890 meters in 10 seconds.
Step-by-step explanation:
Given : 178 meters in 2 seconds.
To find : How many meters in 10 seconds ?
Solution :
Applying unitary method,
In 2 seconds there is 178 meter.
In 1 second there is [tex]\frac{178}{2}[/tex] meter.
In 1 second there is 89 meter.
In 10 seconds there is [tex]10\times 89[/tex] meter.
In 10 seconds there is 890 meter.
Therefore, 178 meters in 2 seconds = 890 meters in 10 seconds.
To supplement your retirement, you estimate that you need to accumulate
$360,000 exactly 38 years from today. You plan to make equal, end-of-year deposits into an account paying 11 % annual interest.
a. How large must the annual deposits be to create the
$360, 000 fund by the end of 38 years?
b. If you can afford to deposit only $590 per year into theaccount, how much will you have accumulated in 38 years?
Answer:
a) $765.13 b) $277,601.23
Step-by-step explanation:
a) The problem is an example of an ordinary annuity (deposits at the end of the period).
The future value of this type of annuity is:
[tex]FV=A*\frac{(1+i)^{n} -1}{i}[/tex]
Clearing the annual deposit A
[tex]A=FV*\frac{i}{(1+i)^{n} -1}[/tex]
[tex]A=360,000*\frac{0.11}{(1.11)^{38}-1 } =360,000*0,002125351=765.13[/tex]
The deposit needed to have $360,000 in 38 years is $765.13
b) We can use the same formula to compute the FV of a known deposit:
[tex]FV=A*\frac{(1+i)^{n} -1}{i}[/tex]
[tex]FV=590*\frac{(1.11)^{38} -1}{0.11}=590*470,5105644=277,601.23[/tex]
With annual deposits of $590 you will have at 38 years an ammount of $277,601.23
The owner of the Rancho Los Feliz has 7000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. Fencing is not required along the river, and the length of the fencing parallel to the river is to exceed the length of the fencing perpendicular to it by 2500 yd. Find the area of the enclosed land (in sq yd).
Answer:
6000000 sq yd
Step-by-step explanation:
Data provided in the question:
Length of the fencing = 7000 yd
let the perpendicular sides be 'P'
and the length parallel to the river be 'L'
according to the given question
L = P + 2500 ............(1)
also,
Length to be fenced = 2P + L
thus,
2P + L = 7000 ...........(2)
substituting L from (1), we get
2P + P + 2500 = 7000
or
3P = 7000 - 2500
or
3P = 4500
or
P = 1500 yd
Thus,
L = 1500 + 2500 = 4000 yd
Therefore,
the area of the rectangular land = L × P = 4000 × 1500 = 6000000 sq yd
Answer:
Area of land = 6000000 sq yd
Step-by-step explanation:
Given,
length of fencing= 7000 yd
Let's assume that the length of the land parallel to the river is l and the breadth of the land perpendicular to the river is b.
Then, it is given that
l = b +2500
Since, there is no need of fencing along the river so, we can write
l +2b = 7000
=>b+2500 = 7000
=> b = 7000-2500
= 4000
As the area of rectangular land can be given as
A = length x breadth
= 4000 x 2500 sq yd
= 6000000 sq yd
So, the area of the enclosed land will be 6000000 sq yd.
A(1,3,5, 7,9}, B {0,1,2,3,4,5,6, 7,8,9} be the universal set and let A (0,2,4,6,8}, and C {2,3,5,7}. List the elements of the following sets 1.) Let U (a) An B (b) C\B (c) AU (Bnc)
Answer:
a) [tex]A \cap B = \{\phi\}[/tex]
b)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
c)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
Step-by-step explanation:
We are given the following information:
The universal set is : U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {0, 2, 4, 6, 8}
B = {1, 3, 5, 7, 9}
C = {2, 3, 5, 7}
a) [tex]A \cap B = \{\phi\}[/tex]
b)[tex] C/B = C \cap \overline{B} = \{2, 3, 5, 7\} \cap \{0, 2, 4, 6, 8\} = \{2\}[/tex]
c)
[tex]A \cup (B \cap C) = \{0, 2, 4, 6, 8\} \cup (\{1, 3, 5, 7, 9\} \cap \{2, 3, 5, 7\})\\=\{0, 2, 4, 6, 8\} \cup \{3, 5, 7\}\\= \{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection. Give your answers in degrees, rounding to one decimal place. Enter your answers as a comma-separated list.) y = 7x2, y = 7x3
Answer:
The angles between the curves at the points of intersection are:
0º, 1.3º
Step-by-step explanation:
The intersections points are found by setting the equations equal to each other and solving the resulting equation:
[tex]7x^2=7x^3\\x^3-x^2=0\\x^2(x-1)=0\\x=0,x=1[/tex]
The angles of the tangent lines can be found by stating their slopes.
To find the slope we differentiate the equations:
[tex]y'_1=14x,y'_2=21x^2[/tex]
Then we plug the x-coordinates of the intersections:
For x=0 we get the slopes are both 0:
[tex]y'_1=14(0)=0,y'_2=21(0)^2=0[/tex]
So the angles of inclination of the lines are the same their difference is 0. Hence the angle between the tangent curves is also 0º at the point of intersection at x=0
For x=1 we get the following slopes:
[tex]y'_1=14(1)=14,y'_2=21(1)^2=21[/tex]
The slopes are the tangents of the angles. Therefore, to get the angle between the lines we do:
[tex]arctan(21)-arctan(14)\approx87.2737\º-85.9144\º\approx1.3\º[/tex]
So, 1.3º is the angle between the curves at the second point of intersection at x=1.
If a population of dolphins increases at a constant rate of 1.5% every year for 20 years what will be the total percentage increase over the 20 years.
Answer:
The population increased by 34.69% over 20 years.
Step-by-step explanation:
It is given that the population of dolphins increases at a constant rate of 1.5% every year for 20 years.
Formula for population increase:
[tex]P=a(1+r)^t[/tex]
where, a is initial population, r is growth rate and t is time in years.
If the population of dolphins increases at a constant rate of 1.5% every year for 20 years, then the population after 20 years is
[tex]P=a(1+0.015)^{20}[/tex]
[tex]P=a(1.015)^{20}[/tex]
[tex]P=1.346855a[/tex]
Where, a is the initial population.
The total percentage increase over the 20 years is
[tex]\% change=\frac{P-a}{a}\times 100[/tex]
where, P is population after 20 years and a is initial amount.
[tex]\% change=\frac{1.346855a-a}{a}\times 100[/tex]
[tex]\% change=\frac{0.346855a}{a}\times 100[/tex]
[tex]\% change=0.346855\times 100[/tex]
[tex]\% change=34.6855[/tex]
[tex]\% change\approx 34.69[/tex]
Therefore the population increased by 34.69% over 20 years.
Convert 500 cubic feet to liters then to gallons
Answer:
500 cubic feet equals 14158.4 liters or 3740.25 gallons.
Step-by-step explanation:
We are asked to convert 500 cubic feet to liters then to gallons.
We know that one cubic feet equals 28.3168 liters.
[tex]\text{500 cubic feet to liters}=\text{500 cubic feet}\times \frac{\text{28.3168 liters}}{\text{cubic feet}}[/tex]
[tex]\text{500 cubic feet to liters}=500\times \text{28.3168 liters}[/tex]
[tex]\text{500 cubic feet to liters}=\text{14158.4 liters}[/tex]
We know one liter equals 0.264172 gallons.
[tex]\text{14158.4 liters to gallons}=\text{14158.4 liters}\times\frac{0.264172\text{ gallons}}{\text{liter}}[/tex]
[tex]\text{14158.4 liters to gallons}=14158.4 \times 0.264172\text{ gallons}[/tex]
[tex]\text{14158.4 liters to gallons}=3740.2528448\text{ gallons}[/tex]
[tex]\text{14158.4 liters to gallons}\approx 3740.25\text{ gallons}[/tex]
Therefore, 500 cubic feet equals 14158.4 liters or 3740.25 gallons.
Convert these decimals into fractions and simplify as much as possible: a) 0.25 b) 0.08 C) 0.400 d) 1.1 e) 3.5
Answer:
When we change a decimal into fraction, then we follow the following steps,
Step 1 : first we write the decimal number with the denominator 1,
Step 2 : Multiply numerator by 10s ( eg 10, 100, 100 etc ) for omitting decimal.
Step 3 : Multiply the denominator ( i.e 1 ) by the same number,
Step 4 : Reduce the fraction in the simplest form if possible by dividing both numerator and denominator by the HCF of numerator and denominator.
a)
[tex]0.25 =\frac{0.25}{1}=\frac{0.25\times 100}{100}=\frac{25}{100}=\frac{25\div 25}{100\div 25}=\frac{1}{4}[/tex]
b)
[tex]0.08 =\frac{0.08}{1}=\frac{0.08\times 100}{100}=\frac{8}{100}=\frac{8\div 4}{100\div 4}=\frac{2}{25}[/tex]
c)
[tex]0.400 =\frac{0.4}{1}=\frac{0.4\times 10}{10}=\frac{4}{10}=\frac{4\div 2}{10\div 2}=\frac{2}{5}[/tex]
d)
[tex]1.1 =\frac{1.1}{1}=\frac{1.1\times 10}{10}=\frac{11}{10}[/tex]
e)
[tex]3.5 =\frac{3.5}{1}=\frac{3.5\times 10}{10}=\frac{35}{10}=\frac{35\div 5}{10\div 5}=\frac{7}{2}[/tex]
Final answer:
To convert decimals into simplified fractions: 0.25 is 1/4, 0.08 is 2/25, 0.400 is 2/5, 1.1 is 11/10 and 3.5 is 7/2. Numbers in scientific notation are written in decimal form by adjusting the decimal point. When rounding to three significant figures, ensure only the first three digits after the leading non-zero digit are kept.
Explanation:
When converting decimals to fractions and simplifying them, it's important to consider the place value of the decimal. Here's how you would convert and simplify the provided decimals:
0.25 can be written as 25/100, which simplifies to 1/4.
0.08 is 8/100, which simplifies to 1/12.5 or 2/25 when expressed as a simplified fraction.
0.400 is 400/1000, which simplifies to 2/5.
1.1 is equivalent to 11/10 or 1 1/10 in mixed number form.
3.5 equals 35/10, which simplifies to 7/2 or 3 1/2 in mixed number form.
For scientific notation, numbers are converted to their decimal forms by moving the decimal point:
5.65 x 10-3 means the decimal point is moved 3 places to the left, giving 0.00565.
9.25 x 10-4 means the decimal point is moved 4 places to the left, resulting in 0.000925.
To write numbers in scientific notation:
4500 becomes 4.5 x 103.
2220000 turns into 2.22 x 106.
0.0035 is 3.5 x 10-3.
0.7 can be written as 7 x 10-1.
858.67 is expressed as 8.5867 x 102.
When rounding to three significant figures:
0.0004505 becomes 4.51 x 10-4 (count starts from the first non-zero digit).
0.00045050 also rounds to 4.51 x 10-4.
For 7.210 x 106, it remains unchanged as it already has three significant figures.
5.00 x 10-6 stays the same, with three significant figures present.
Convert 17.42 m to customary units. A.57'-17/8" B. 36-10 1/2" C. 442 1/2" D. 367/8" E. None of these answers is reasonable.
Answer:
Option E - None of these answers is reasonable.
Step-by-step explanation:
To find : Convert 17.42 m to customary units ?
Solution :
The customary units is defined as the measure length and distances in the customary system are inches, feet, yards, and miles.
The options belong to feet and inches.
We have to convert meter into inches, feet.
Meter into feet,
[tex]1 \text{ feet} = 0.3048 \text{ meter}[/tex]
[tex]1 \text{ meter} = \frac{1}{0.3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} = \frac{17.42}{0.3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} = \frac{174200}{3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} =57 \frac{464}{3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} =57 \frac{58}{381}\text{ feet}[/tex]
Now, Feet into inches
[tex]1 \text{ feet} = 12\text{ inches}[/tex]
[tex] \frac{58}{381} \text{ feet} = 12\times \frac{58}{381}\text{ inches}[/tex]
[tex] \frac{58}{381} \text{ feet} =\frac{232}{381}\text{ inches}[/tex]
i.e. [tex]17.42 \text{ meter} =57\text{ feet }\frac{232}{381}\text{ inches}[/tex]
or [tex]17.42 \text{ meter} =57'\frac{232}{381}''[/tex]
None of these answers is reasonable.
Therefore, Option E is correct.
In a recent poll, 850 people were surveyed. 69% of them said they opposed the current war. How many people stated they were in opposition to the war?
Answer:
586 people stated that they were in opposition to the war.
Step-by-step explanation:
Percentage problems can be solved as a simple 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. In this case, the rule of three is a cross multiplication.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.
A percentage problem is an example where the relationship between the measures is direct.
The problem states that 69% of the said they opposed the current war. 69% is 69 of 100. How much it is of 850.
So
69 - 100
x - 850
100x = 69*850
[tex]x = \frac{58650}{100}[/tex]
x = 586.5
586 people stated that they were in opposition to the war.
Let A and B be two events in a sample space S such that P(A) = 0.5, P(B) = 0.6, and P(A intersectionB) = 0.15. Find the probabilities below. Hint: (A intersectionBc) union (A intersectionB) = A.
(a) P(A|Bc)
(b) P(B|Ac)
Answer:
(a) [tex]\frac{7}{8}[/tex]
(b) [tex]\frac{9}{10}[/tex]
Step-by-step explanation:
Given,
P(A) = 0.5 ⇒ [tex]P(A^c)=1-P(A) = 1 - 0.5 = 0.5[/tex]
P(B) = 0.6 ⇒ [tex]P(B^c)=1-P(B) = 1 - 0.6 = 0.4[/tex]
P(A∩B) = 0.15
∵ [tex]P(A\cap B^c)=P(A) - P(A\cap B) = 0.5 - 0.15 = 0.35[/tex]
Similarly,
[tex]P(B\cap A^c)=P(B) - P(B\cap A) = 0.6 - 0.15 = 0.45[/tex]
Now,
(a) [tex]P(\frac{A}{B^c})=\frac{P(A\cap B^c)}{P(B^c)}=\frac{0.35}{0.4}=\frac{35}{40}=\frac{7}{8}[/tex]
(b) [tex]P(\frac{B}{A^c})=\frac{P(B\cap A^c)}{P(A^c)}=\frac{0.45}{0.5}=\frac{45}{50}=\frac{9}{10}[/tex]
graph the function f (x)=x4 -7x2 on the interval x=-3 to x=3 which of the following best describes the graph from left to right
Answer:
when graphing f(x) between x=-3 & x=3, the result is a function that comes from x=-infinite and positive y, crosses the x-axis at (-2.646,0), continues to decrease until (-1.871,-12.25) and then increases until (0,0).
This function is symetrical by the y axis, therefore, after reaching (0,0), f(x) decreases until (1.871,-12.25), starts to increase until it crosses the x axis at (2-646,0) and continues to increase until x=+infinite
Step-by-step explanation:
This funcion appears as a large W, with it's points on (-1.871,-12.25) , (0,0) & (1.871,-12.25)
Using data collected between 1957 and 1978, from 15 samples of adults, Professor Rodgers (1982 Social Forces) found that (1) the average level of happiness reported by people under 65 years of age declined from 1957 to 1970. For this same group, the average level of happiness increased slightly from 1970 to 1978. (2) The average level of happiness reported by people age 65 and older increased from 1957 to 1978. A (the) variable(s) in this study is (are)
Answer:
The variable in this study is age.
Step-by-step explanation:
The variable in this study is Age, which has a relationship of cause and effect. Consequently,it is clear that happiness does not depend on the passing of time , but on the age of each group of people.
U.S. postal regulations require that a package can have a maximum combined length and girth (distance around) of 108 in. A shipping carton is constructed so that it has a width of 14 in., a height of 12 in., and can be cut or folded to various lengths. What is the maximum length that can be used?
To determine the maximum length of the shipping carton given a maximum combined length and girth of 108 inches, we calculate the girth as 52 inches using the provided width and height. Subtracting this from 108 inches gives us a maximum allowable length of 56 inches.
The problem is a geometry optimization problem where we need to maximize the length of a shipping carton given a constraint on its combined length and girth. Girth is the perimeter around the width and height sides of the package. According to the U.S. postal regulations, the maximum combined length and girth is 108 inches.
First, let's calculate the girth using the given dimensions of the carton. The width is 14 inches and the height is 12 inches. The girth is twice the width plus twice the height (since girth is the perimeter of the cross-section):
girth = 2 × width + 2 × height
girth = 2 × 14in + 2 × 12in
girth = 28in + 24in
girth = 52in
Now, to find the maximum length, we can subtract the girth from the maximum allowed combined length and girth:
max length = max combined length and girth - girth
max length = 108in - 52in
max length = 56in
Therefore, the maximum length that the carton can be is 56 inches.
Gianna is going to throw a ball from the top floor of her middle school. When she throws the hall from 48 feet above the ground, the function h(t)=-16t^2+32t+48 models the height,h, of the ball above the ground as a function of time,t. Find the times the ball will be 48 feet above the ground.
Answer:
So, the times the ball will be 48 feet above the ground are t = 0 and t = 2.
Step-by-step explanation:
The height h of the ball is modeled by the following equation
[tex]h(t)=-16t^2+32t+48[/tex]
The problem want you to find the times the ball will be 48 feet above the ground.
It is going to be when:
[tex]h(t) = 48[/tex]
[tex]h(t)=-16t^{2}+32t+48[/tex]
[tex]48=-16t^{2}+32t+48[/tex]
[tex]0=-16t^{2}+32t+48 - 48[/tex]
[tex]16t^{2} - 32t = 0[/tex]
We can simplify by 16t. So
[tex]16t(t-2)= 0[/tex]
It means that
16t = 0
t = 0
or
t - 2 = 0
t = 2
So, the times the ball will be 48 feet above the ground are t = 0 and t = 2.
A recipe calls for 2 tsp of salt, 1 tsp of pepper, and 4 tsp of garlic powder. How much pepper and garlic powder would you need for a larger batch if you are using 15 tsp of salt?
Answer:
amount of pepper required= 7.5 tsp
amount of garlic powder required = 30 tsp
Step-by-step explanation:
Given,
amount of salt used for small batch of the recipe = 2 tsp
amount of pepper used for small batch of the recipe = 1 tsp
amount of garlic powder used for small batch of the recipe = 4 tsp
amount of salt used for the larger batch = 15 tsp
= 2 x 7.5 tsp
= amount of salt used for small batch the recipe x 7.5
So,
the amount of pepper needed for the larger batch= 7.5 x amount of pepper used for the small batch of recipe
= 7.5 x 1 tsp
= 7.5 tsp
the amount of garlic powder needed for the larger batch= 7.5 x amount of garlic powder used for the small batch of recipe
= 7.5 x 4 tsp
= 30 tsp
Final answer:
To adjust the recipe for 15 tsp of salt, you will need 7.5 tsp of pepper and 30 tsp of garlic powder, by applying a scaling factor based on the original recipe proportions.
Explanation:
The question asks how much pepper and garlic powder are needed if a recipe is scaled up to use 15 tsp of salt, from an original recipe that calls for 2 tsp of salt, 1 tsp of pepper, and 4 tsp of garlic powder. To solve this, we first determine the scaling factor for the recipe by dividing the new quantity of salt by the original quantity of salt, which is 15 tsp ÷ 2 tsp = 7.5. Next, we apply this scaling factor to the measurements for pepper and garlic powder.
Pepper needed = 1 tsp (original amount) x 7.5 (scaling factor) = 7.5 tsp of pepper.
Garlic Powder needed = 4 tsp (original amount) x 7.5 (scaling factor) = 30 tsp of garlic powder.
Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent?
Answer:
the probability that the user is fraudulent is 0.00299133
Step-by-step explanation:
Let be the events be:
G: The user generates calls from two or more areas.
NG: The user does NOT generate calls from two or more areas.
L: The user is legitimate.
F: The user is fraudulent.
The probabilities established in the statement are:
[tex]P (G | L) = 0.01//P (G | F) = 0.30//P (F) = 0.0001//P (L) = 0.9999//[/tex]
With these values, the probability that a user is fraudulent, if it has originated calls from two or more areas is:
[tex]P (F|G) = \frac{P(F\bigcap G)}{P(G)} = \frac{P(F)P(G|F)}{P(G)} = \frac{P(F)P(G|F)}{P(F)P(G|F)+P(L)P(G|L)}[/tex]
[tex]\frac{(0.0001)(0.30)}{(0.0001)(0.30)+(0.9999)(0.01)} = 0.00299133[/tex]
linear equations: 1.5x + 1.3x= -8.4
1.5x + 1.3x = - 8.4
2.8x = -8.4
x= - 8.4/2.8
x = 3
YOURE WELCOME PLEASE MARK ME AS BRAINLIEST AND FOLLOW ME ❤️❤️❤️❤️❤️ AU REVOIROn simplification of liner equation 1.5x + 1.3x = -8, we get x = -3.
To solve the linear equation 1.5x + 1.3x = -8.4, we need to start by combining like terms.
Both terms on the left side of the equation have the variable x, so we can add them together.
1.5x + 1.3x = 2.8x.
2.8x = -8.4.
To find the value of x, we need to isolate the variable by dividing both sides of the equation by 2.8.
Divide both sides by 2.8:
x = -8.4 / 2.8.
x = -3.
To complete your spring schedule, you must add Calculus and Physics. At 9:30, there are three Calculus sections and two Physics sections; while at 11:30, there are two Calculus sections and three Physics sections. How many ways can you complete your schedule if your only open periods are 9:30 and 11:30?
Answer:
You can complete your schedule in 13 different ways.
Step-by-step explanation:
You initially have two general options:
1) You can take Calculus at 9.30 and Physics at 11.30 OR
2) You can take Physics at 9.30 and Calculus at 11.30.
Let's examine each option:
1) If you take Calculus at 9.30 you'd have 3 options (since there are 3 Calculus sections), and then you'd have 3 options at 11.30 to take Physics. This makes 3 x 3 = 9 options.
2) If you choose to take Physics at 9.30, you'd have 2 options and then you'd have 2 Calculus options at 11.30. This makes 2x2 = 4 options.
Since you can take either option one OR two, we will sum up both results, and therefore you have 9 +4 = 13 different ways to complete your schedule.