[tex]h(x)=(5-2x^6)^3+\dfrac1{5-2x^6}[/tex]
Let [tex]g(x)=5-2x^6[/tex] and [tex]f(x)=x^3+\dfrac1x[/tex]. Then [tex]h(x)=f(g(x))[/tex].
Set [tex]u=5-2x^6[/tex]. By the chain rule,
[tex]\dfrac{\mathrm dh}{\mathrm dx}=\dfrac{\mathrm dh}{\mathrm du}\cdot\dfrac{\mathrm du}{\mathrm dx}[/tex]
Since [tex]h(u)=u^3+\dfrac1u[/tex] and [tex]u(x)=5-2x^6[/tex], we have
[tex]\dfrac{\mathrm dh}{\mathrm du}=3u^2-\dfrac1{u^2}[/tex]
[tex]\dfrac{\mathrm du}{\mathrm dx}=-12x^5[/tex]
Then
[tex]\dfrac{\mathrm dh}{\mathrm dx}=\left(3u^2-\dfrac1{u^2}\right)(-12x^5)=\boxed{-12x^5\left(3(5-2x^6)^2-\dfrac1{(5-2x^6)^2}\right)}[/tex]
which we could rewrite slightly as
[tex]\dfrac{\mathrm dh}{\mathrm dx}=-\dfrac{12x^5(3(5-2x^6)^4-1)}{(5-2x^6)^2}[/tex]
To differentiate the given function using the chain rule, we need to identify the functions f(x) and g(x), then calculate their derivatives. Once we have the derivatives, we can apply the chain rule to find the derivative of the composite function h(x).
Explanation:Chain Rule
To differentiate the function h(x) = (5 - 2x^6)³ + 1/(5 - 2x^6) using the chain rule, we can write it as the composite function h(x) = f(g(x)).
Let's identify the functions f(x) and g(x):
f(x) = x³, g(x) = (5 - 2x^6)
Next, let's calculate the derivatives of f(x) and g(x):
f'(x) = 3x², g'(x) = -12x^5
Finally, we can apply the chain rule to differentiate h(x):
h'(x) = f'(g(x)) * g'(x) = (3(5 - 2x^6)²) * (-12x^5)
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A square pyramid is 6 feet on each side. The height of the pyramid is 4 feet. What is the total area of the pyramid?
60 ft2
156 ft2
96 ft2
120 ft2
Answer:
Option C
Step-by-step explanation:
96ft2
Answer:
Area of pyramid = [tex]96[/tex]. square feet.
Step-by-step explanation:
Given : A square pyramid is 6 feet on each side. The height of the pyramid is 4 feet.
To find: What is the total area of the pyramid.
Solution : We have given
Each side of square pyramid = 6 feet .
Height = 4 feet .
Area of pyramid = [tex](side)^{2} + 2* side\sqrt{\frac{(side)^{2}}{4} +height^{2}}[/tex].
Plug the values side = 6 feet , height = 4 feet .
Area of pyramid = [tex](6)^{2} + 2* 6\sqrt{\frac{(6)^{2}}{4} + 4^{2}}[/tex].
Area of pyramid = [tex]36+ 12\sqrt{\frac{36}{4} + 16}[/tex].
Area of pyramid = [tex]36+ 12\sqrt{9 +16}[/tex].
Area of pyramid = [tex]36+ 12\sqrt{25}[/tex].
Area of pyramid = [tex]36+ 12 *5[/tex].
Area of pyramid = [tex]36+ 60[/tex].
Area of pyramid = [tex]96[/tex]. square feet.
Therefore, Area of pyramid = [tex]96[/tex]. square feet.
A student answers a multiple-choice examination question that offers four possible answers. Suppose the probability that the student knows the answer to the question is 0.9 and the probability that the student will guess is 0.1. Assume that if the student guesses, the probability of selecting the correct answer is 0.25. If the student correctly answers a question, what is the probability that the student really knew the correct answer? (Round your answer to four decimal places.)
Answer: 0.9730
Step-by-step explanation:
Let A be the event of the answer being correct and B be the event of the knew the answer.
Given: [tex]P(A)=0.9[/tex]
[tex]P(A^c)=0.1[/tex]
[tex]P(B|A^{C})=0.25[/tex]
If it is given that the answer is correct , then the probability that he guess the answer [tex]P(B|A)= 1[/tex]
By Bayes theorem , we have
[tex]P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(C|A^c)P(A^c)}[/tex]
[tex] =\dfrac{(1)(0.9)}{(1))(0.9)+(0.25)(0.1)}\\\\=0.972972972973\approx0.9730[/tex]
Hence, the student correctly answers a question, the probability that the student really knew the correct answer is 0.9730.
Expand the logarithm log 7(3x − 2)^2
[tex]\textbf{Transform}\\ \textrm{log} (7(3x -2)^2) \textbf{ into} \textrm{ log}(7) + \textrm{log}(3x-2)^2\\\\ \textbf{Expand} \\ \text{log}(3x-2)^2\\\\ \text{You can move 2 outside of }\text{log}(3x-2)^2\\\\ \textbf{Answer}\\ \text{log }7 + 2\text{ log}(3x-2)[/tex]
Two friends went to get ice cream sundaes. They each chose a flavor of ice cream from a list of vanilla and chocolate and toppings from a list of hot fudge, strawberries, sprinkles, peanuts, and whipped cream. Use the sets below describing their choices and find B'.
Let A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
Let B = {vanilla, hot fudge, sprinkles, whipped cream}
Let C = {chocolate, hot fudge, peanuts, whipped cream}
{vanilla, hot fudge, sprinkles, whipped cream}
{chocolate, strawberries, peanuts}
{chocolate, strawberries}
{vanilla, strawberries, peanuts}
Answer:
{chocolate, strawberries, peanuts}
Step-by-step explanation:
Given that three sets are
Let A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
Let B = {vanilla, hot fudge, sprinkles, whipped cream}
Let C = {chocolate, hot fudge, peanuts, whipped cream}
Then Universal set U = AUBUC
= {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
B'=elements in U but not in B
={chocolate, strawberries, peanuts}
The resulting set is B' = {chocolate, strawberries, peanuts}.
To solve for B', we first need to understand that B' (B complement) consists of elements that are in set A but not in set B.
Given the sets:
A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}B = {vanilla, hot fudge, sprinkles, whipped cream}C = {chocolate, hot fudge, peanuts, whipped cream}Set B includes: vanilla, hot fudge, sprinkles, and whipped cream. Therefore, B' will be the elements of set A excluding those in B.
Thus, B' is:
chocolatestrawberriespeanutsTherefore, the set B' = {chocolate, strawberries, peanuts}.
This method can help you understand combinations without repetition effectively.
You pick 7 digits (0-9) at random without replacement, and write them in the order picked. What is the probability that you have written the first 7 digits of your phone number
The probability of writing the first 7 digits of your phone number is 1/60480.
Explanation:To determine the probability of choosing the first 7 digits of your phone number in the given scenario, we need to calculate the probability of choosing each digit correctly and in order. Since there are 10 digits to choose from, the probability of choosing the first digit correctly is 1/10. The probability of choosing the second digit correctly is 1/9, since one digit has already been chosen. Continuing this pattern, the probability of choosing all 7 digits correctly and in order is:
P(choosing all seven numbers correctly) = P(choosing 1st number correctly) * P(choosing 2nd number correctly) * ... * P(choosing 7th number correctly)
So, the probability is:
1/10 * 1/9 * 1/8 * 1/7 * 1/6 * 1/5 * 1/4 = 1/60480
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The probability of writing the first 7 digits of your phone number is 1/604,800.
Explanation:The probability of writing the first 7 digits of your phone number depends on the specific digits in your phone number. However, assuming that all digits are equally likely to be chosen, the probability can be calculated by multiplying the probabilities of choosing each digit correctly. Since there are 10 digits to choose from and you are picking 7, the probability would be:
Probability of choosing the first digit correctly: 1/10Probability of choosing the second digit correctly: 1/9 (since you are picking without replacement)Probability of choosing the third digit correctly: 1/8Probability of choosing the fourth digit correctly: 1/7Probability of choosing the fifth digit correctly: 1/6Probability of choosing the sixth digit correctly: 1/5Probability of choosing the seventh digit correctly: 1/4To calculate the overall probability, you multiply these individual probabilities together:
1/10 * 1/9 * 1/8 * 1/7 * 1/6 * 1/5 * 1/4= 1/(10*9*8*7*6*5*4)= 1/604,800So, the probability of writing the first 7 digits of your phone number is 1/604,800.
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The AWP for a gallon (3785 ml) of antihistamine/ antitussive cough syrup is $18.75, with an additional 20% discount from the wholesaler. What is the cost of 1 pint of the medication?
Answer:
The cost of 1 pint of the medication would be $1.875.
Step-by-step explanation:
The AWP of 3785 ml ( 1 gallon ) cough syrup = $18.75
After an additional 20% discount from wholesaler the price would be
New price = 18.75 - (0.20 × 18.75)
= 18.75 - 3.75
= $15.00
Since 1 gallon ( 3785 ml) = 8 pints
Therefore, the price for 1 pint = [tex]\frac{15}{8}[/tex] = $1.875
The cost of 1 pint of the medication would be $1.875.
The probability that a student graduating from Suburban State University has student loans to pay off after graduation is .60. If two students are randomly selected from this university, what is the probability that neither of them has student loans to pay off after graduation?
Answer: 0.16
Step-by-step explanation:
Given: The probability that a student graduating from Suburban State University has student loans to pay off after graduation is =0.60
Then the probability that a student graduating from Suburban State University does not have student loans to pay off after graduation is =[tex]1-0.6=0.4[/tex]
Since all the given event is independent for all students.
Then , the probability that neither of them has student loans to pay off after graduation is given by :-
[tex](0.4)\times(0.4)=0.16[/tex]
Hence, the probability that neither of them has student loans to pay off after graduation =0.16
Define a function sinc(x) (pronounced "sink of x") by: sinc(x)= {sin(x)/x if x is not 0 {1 if x = 0 Use this list of Basic Taylor Series to find the Taylor Series for f(x) = (sinc(x)) based at 0. a.Give your answer using summation notation. b.Give the interval on which the series converges.
You probably know that
[tex]\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}[/tex]
Then
[tex]\mathrm{sinc}\,x=\displaystyle\frac1x\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}[/tex]
when [tex]x\neq0[/tex], and 1 when [tex]x=0[/tex].
By the ratio test, the series converges if the following limit is less than 1:
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+2}}{(2n+3)!}}{\frac{(-1)^nx^{2n}}{(2n+1)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)!}{(2n+3)!}[/tex]
The limit is 0, so the series converges for all [tex]x[/tex].
If f(x)=3x^2-2 and g(x)=4x+2, what is the value of (f+g)(2)
[tex](f+g)(x)=3x^2-2+4x+2=3x^2+4x\\\\(f+g)(2)=3\cdot2^2+4\cdot2=12+8=20[/tex]
Walgreens sells both Kleenex and NyQuil. In a random sample of 5000 purchases, it was found that 1200 contained Kleenex and 540 contained NyQuil. These are not independent because sick people are more likely to buy both and healthy people more likely to buy neither. In looking into that, it was found that 360 purchases contained both.
a. Make a two-way table to display this information. Fill in the table with the missing entries, and be sure your table has row and column totals.
b. If we randomly select a purchase that contained Kleenex, what is the probability it also contained NyQuil?
c. What percent of all purchases didn’t contain either product?
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The Bailey family and the Harris family each used their sprinklers last summer. The water output rate for the Bailey family's sprinkler was 15 L per hour. The water output rate for the Harris family's sprinkler was 40 L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 1325
. How long was each sprinkler used?
Answer:
Bailey family's sprinkler was used for 35 hours and Harris family's sprinkler was used for 20 hours.
Step-by-step explanation:
Set up a system of equations.
Let be "b" the time Bailey family's sprinkler was used and "h" the time Harris family's sprinkler was used.
Then:
[tex]\left \{ {{b+h=55} \atop {15b+40h= 1,325}} \right.[/tex]
You can use the Elimination method. Multiply the first equation by -15, then add both equations and solve for "h":
[tex]\left \{ {{-15b-15h=-825} \atop {15b+40h= 1,325}} \right.\\.............................\\25h=500\\\\h=\frac{500}{25}\\\\h=20[/tex]
Substitute [tex]h=20[/tex] into an original equation and solve for "b":
[tex]b+20=55\\\\b=55-20\\\\b=35[/tex]
Problem Page
A delivery truck is transporting boxes of two sizes: large and small. The combined weight of a large box and a small box is 80 pounds. The truck is transporting 55 large boxes and
70 small boxes. If the truck is carrying a total of 4850 pounds in boxes, how much does each type of box weigh?
For this case we propose a system of equations:
x: Variable representing the weight of large boxes
y: Variable that represents the weight of the small boxes
So
[tex]x + y = 80\\55x + 70y = 4850[/tex]
We clear x from the first equation:
[tex]x = 80-y[/tex]
We substitute in the second equation:
[tex]55 (80-y) + 70y = 4850\\4400-55y + 70y = 4850\\15y = 450\\y = 30[/tex]
We look for the value of x:
[tex]x = 80-30\\x = 50[/tex]
Large boxes weigh 50 pounds and small boxes weigh 30 pounds
Answer:
Large boxes weigh 50 pounds and small boxes weigh 30 pounds
Answer: A large box weighs 50 pounds and a small box weighs 30 pounds.
Step-by-step explanation:
Set up a system of equations.
Let be "l" the weight of a large box and "s" the weight of a small box.
Then:
[tex]\left \{ {{l+s=80} \atop {55l+70s=4,850}} \right.[/tex]
You can use the Elimination method. Multiply the first equation by -55, then add both equations and solve for "s":
[tex]\left \{ {{-55l-55s=-4,400} \atop {55l+70s=4,850}} \right.\\.............................\\15s=450\\\\s=\frac{450}{15}\\\\s=30[/tex]
Substitute [tex]s=30[/tex] into an original equation and solve for "l":
[tex]l+(30)=80\\\\l=80-30\\\\l=50[/tex]
A theater group made appearances in two cities. The hotel charge before tax in the second city was $500 lower than in the first. The tax in the first city was 6.5% and the tax in the second city was 4.5% The total hotel tax paid for the two cities was $582.50
. How much was the hotel charge in each city before tax?
Answer:
First city: $5,500
Second city: $5,000
Step-by-step explanation:
Let's define x as the hotel price in the first city and y the hotel price in the second city. We can start with this equation:
y = x - 500 (The hotel before tax in the 2nd city was $500 lower than in the 1st.)
Then we can say
0.065x + 0.045y = 582.50 (the sum of the tax amounts were $582.50)
We place the value of y from the first equation in the second equation:
0.065x + 0.045 (x - 500) = 582.50
0.065x + 0.045x - 22.50 = 582.50 (simplifying and adding 22.5 on each side)...
0.11x = 605
x = 5,500
The cost of the first hotel was $5,500
Thus, the cost of the second hotel was $5,000 (x - 500)
Given the stem and leaf plot, which of the following statements is true?
STEM LEAF
2 9
3 2 6 7
4 1 2
5 0
6
7 5
a) There are no outliers on the stem and leaf plot; b) the numbers 29 and 75 are the outliers on the stem and leaf plot; c) the number 75 is an outlier on the stem and leaf plot; d) the number 60 is the outlier on the stem and leaf plot.
Answer: c) The number 75 is an outlier on the stem and leaf plot.
Step-by-step explanation:
An outlier is a value in data which is extremely large or small from all the values. It appears detached from the other values in data.When we look in the given stem-leaf plot, there is no leaf attached to the stem with value 6.
It mean there is no value between 50 and 75.
It shows that the value of 75 is detached from the other values in the data.
⇒ The number 75 is an outlier on the stem and leaf plot.
Answer:
The answer is C
Step-by-step explanation:
Combine the following expressions.
Answer right pls, thanks
a and c both have √x , so they will both be in brackets multiplied by √x.
b is the only term with √y so it will be outside of the brackets.
So the answer will be:
(a - c)√x + b√y
We can check this by expanding the brackets:
(a - c)√x + b√y
= a√x - c√x + b√y
We can rearrange this to get the same original expression:
a√x - c√x + b√y
= a√x + b√y - c√x
____________________________________
Answer:
Last option: (a - c)√x + b√y
Answer:
choice 3 is correct √x(a - c) + b√y
explanation:
You simplify by looking for the common multiplier which is √x
meaning it will be
√x(a - c) + b√y
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f(t) = sqrt 9-x
[tex]f'(x)= \dfrac{-1}{2\sqrt{9-x}}[/tex]
The domain of the function is: [tex]x\leq 9[/tex]and the domain of the derivative function is: [tex]x\leq 9[/tex]Step-by-step explanation:The function f(x) is given by:
[tex]f(x)=\sqrt{9-x}[/tex]
The domain of the function is the possible values of x where the function is defined.
We know that the square root function [tex]\sqrt{x}[/tex] is defined when x≥0.
Hence, [tex]\sqrt{9-x}[/tex] will be defined when [tex]9-x\geq 0\\\\i.e.\\\\x\leq 9[/tex]
Hence, the domain of the function f(x) is: [tex]x\leq 9[/tex]
Also, the definition of derivative of x is given by:
[tex]f'(x)= \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}[/tex]
Hence, here by putting the value of the function we get:
[tex]f'(x)= \lim_{h \to 0} \dfrac{\sqrt{9-(x+h)}-\sqrt{9-x}}{h}\\\\i.e.\\\\f'(x)= \lim_{h \to 0} \dfrac{\sqrt{9-(x+h)}-\sqrt{9-x}}{h}\times \dfrac{\sqrt{9-(x+h)}+\sqrt{9-x}}{\sqrt{9-(x+h)}+\sqrt{9-x}}\\\\\\f'(x)= \lim_{h \to 0} \dfrac{(\sqrt{9-(x+h)}-\sqrt{9-x})(\sqrt{9-(x+h)}+\sqrt{9-x})}{(\sqrt{9-(x+h)}+\sqrt{9-x})\times h}\\\\\\f'(x)= \lim_{h \to 0} \dfrac{9-(x+h)-(9-x)}{(\sqrt{9-(x+h)}+\sqrt{9-x})\times h}[/tex]
Since,
[tex](a-b)(a+b)=a^2-b^2[/tex]
Hence, we have:
[tex]f'(x)= \lim_{h \to 0} \dfrac{-h}{(\sqrt{9-(x+h)}+\sqrt{9-x})\times h}\\\\\\f'(x)= \lim_{h \to 0} \dfrac{-1}{(\sqrt{9-(x+h)}+\sqrt{9-x})}\\\\\\i.e.\\\\\\f'(x)= \dfrac{-1}{2\sqrt{9-x}}[/tex]
Since, the domain of the derivative function is equal to the derivative of the square root function.
Also, the domain of the square root function is: [tex]x\leq 9[/tex]
Hence, domain of the derivative function is: [tex]x\leq 9[/tex]
Answer:
-1/sqrt(1-9x)
Step-by-step explanation:
This is the answer
The lengths of plate glass parts are measured to the nearest tenth of a millimeter. The lengths are uniformly distributed with values at every tenth of a millimeter starting at 590.2, and continuing through 590.8. Determine the mean and variance of the lengths. (a) mean (in tenths of millimeters) Round your answer to two decimal places (e.g. 98.76). (b) variance (in tenths of millimeters2) Round your answer to three decimal places (e.g. 98.765).
Answer: [tex]\text{Mean length}=590.5\ mm\\\\\text{Variance of the lengths}=0.03\ mm[/tex]
Step-by-step explanation:
The mean and variance of a continuous uniform distribution function with parameters m and n is given by :-
[tex]\text{Mean=}\dfrac{m+n}{2}\\\\\text{Variance}=\dfrac{(n-m)^2}{12}[/tex]
Given : [tex] m=590.2\ \ \ n=590.80[/tex]
[tex]\text{Then, Mean=}\dfrac{590.2+590.8}{2}=590.5\ mm\\\\\text{Variance}=\dfrac{(590.8-590.2)^2}{12}=0.03\ mm[/tex]
trains A and B are 240 miles apart. Both start at the same time and travel toward each other. They meet 3 hours later. The speed of train A is 20 miles faster than train B. Find the speed of each train.
Final answer:
Train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.
Explanation:
Let's say the speed of train B is x miles per hour. Since train A is 20 miles per hour faster, the speed of train A is x + 20 miles per hour.
Distance = Speed x Time
For train A, Distance = (x +20) * 3
For train B, Distance = x * 3
Since they meet 240 miles apart, the sum of their distances is 240:
(x + 20) * 3 + x * 3 = 240
3x + 60 + 3x = 240
6x = 180
x = 30
Hence, train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.
All Seasons Plumbing has two service trucks that frequently need repair. If the probability the first truck is available is .73, the probability the second truck is available is .59, and the probability that both trucks are available is .43: What is the probability neither truck is available
Answer: .11
Step-by-step explanation:
Let F be the event that the first truck is available and S be the event that the second truck is available.
The probability of neither truck being available is expressed as P([tex]F^{C}[/tex]∩[tex]S^{C}[/tex]) , where P([tex]F^{C}[/tex]) is the probability that the event F doesn't happen and P([tex]S^{C}[/tex]) is the probability that the event S doesn't happen.
P([tex]F^{C}[/tex])= 1-P(F) = 1-0.73 = 0.27
P([tex]S^{C}[/tex])=1-P(S) = 1-0.59 = 0.41
Since [tex]F^{C}[/tex] and [tex]S^{C}[/tex] aren't mutually exclusive events, then:
P([tex]F^{C}[/tex]∪[tex]S^{C}[/tex]) = P([tex]F^{C}[/tex]) + P([tex]S^{C}[/tex]) - P([tex]F^{C}[/tex]∩[tex]S^{C}[/tex])
Isolating the probability that interests us:
P([tex]F^{C}[/tex]∩[tex]S^{C}[/tex])= P([tex]F^{C}[/tex]) + P([tex]S^{C}[/tex])- P([tex]F^{C}[/tex]∪[tex]S^{C}[/tex])
Where P([tex]F^{C}[/tex]∪[tex]S^{C}[/tex]) = 1 - 0.43 = 0.57
Finally:
P([tex]F^{C}[/tex]∩[tex]S^{C}[/tex]) = 0.27+ 0.41 - 0.57 = 0.11
a(12) = 50- 1.25x
how do I solve it
Answer:
a(12) = 35
Step-by-step explanation:
Given
a(12) = 50- 1.25x
Value of x is 12
50 - 1.25(12)
Simplify
50 - 15
Solve
a(12) = 50 - 15
a(12) = 35
Suppose that we have a sample space S = {E 1, E 2, E 3, E 4, E 5, E 6, E 7}, where E 1, E 2, ..., E 7 denote the sample points. The following probability assignments apply: P(E 1) = 0.1, P(E 2) = 0.15, P(E 3) = 0.15, P(E 4) = 0.2, P(E 5) = 0.1, P(E 6) = 0.05, and P(E 7) = 0.25.
A= {E1, E4, E6}
B= {E2, E4, E7}
C= {E2, E3, E5, E7}
(a) Find P(A), P(B), and P(C). (b) What is P(A ∩ B)? (c) What is P(A ∪ B)? (d) Are events A and C mutually exclusive?
Answer:
(a) The probability of P(A), P(B), and P(C) are 0.35, 0.6 and 0.65 respectively.
(b) The probability of P(A ∩ B) is 0.2.
(c) The probability of P(A ∪ B) is 0.75.
(d) Events A and C mutually exclusive because the intersection of set A and C is null set or ∅.
Step-by-step explanation:
The given sample space is
[tex]S=\{E_1,E_2,E_3,E_4,E_5,E_6,E_7\}[/tex]
[tex]P(E_1)=0.1, P(E_2)=0.15,P(E_3)=0.15,P(E_4)=0.2,P(E_5)=0.1,P(E_6)=0.05, P(E_7)=0.25[/tex]
It is given that
[tex]A=\{E_1,E_4,E_6\}[/tex]
[tex]B=\{E_2,E_4,E_7\}[/tex]
[tex]C=\{E_2,E_3,E_5,E_7\}[/tex]
(a)
[tex]P(A)=P(E_1)+P(E_4)+P(E_6)=0.1+0.2+0.05=0.35[/tex]
[tex]P(B)=P(E_2)+P(E_4)+P(E_7)=0.15+0.2+0.25=0.6[/tex]
[tex]P(C)=P(E_2)+P(E_3)+P(E_5)+P(E_7)=0.15+0.15+0.1+0.25=0.65[/tex]
Therefore the probability of P(A), P(B), and P(C) are 0.35, 0.6 and 0.65 respectively.
(b)
A ∩ B represent the common elements of set A and set B.
[tex]A\cap B=\{E_4\}[/tex]
[tex]P(A\cap B)=P(E_4)=0.2[/tex]
The probability of P(A ∩ B) is 0.2.
(c)
A ∪ B represent all the elements of set A and set B.
[tex]A\cup B=\{E_1,E_2,E_4,E_6,E_7\}[/tex]
[tex]P(A\cup B)=P(E_1)+P(E_2)+P(E_4)+P(E_6)+P(E_7)[/tex]
[tex]P(A\cup B)=0.1+0.15+0.2+0.05+0.25=0.75[/tex]
The probability of P(A ∪ B) is 0.75.
(d)
Set A and C has no common element. So, the intersection of set A and C is empty set.
Yes, events A and C mutually exclusive because the intersection of set A and C is null set or ∅.
The probability of events A, B, and C are calculated by summing the individual probabilities of their constituent sample points. The probability of the intersection of events A and B is equal to the probability of the common sample point. The probability of the union of events A and B is obtained by subtracting the probability of the intersection from the sum of their individual probabilities. Events A and C are not mutually exclusive because they have common sample points.
Explanation:(a) Probability of events A, B, and C:
P(A) = P(E1) + P(E4) + P(E6) = 0.1 + 0.2 + 0.05 = 0.35P(B) = P(E2) + P(E4) + P(E7) = 0.15 + 0.2 + 0.25 = 0.6P(C) = P(E2) + P(E3) + P(E5) + P(E7) = 0.15 + 0.15 + 0.1 + 0.25 = 0.65(b) Probability of intersection of events A and B:
P(A ∩ B) = P(E4) = 0.2
(c) Probability of union of events A and B:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.35 + 0.6 - 0.2 = 0.75
(d) Mutually exclusive events A and C:
No, events A and C are not mutually exclusive because they have common sample points in E2 and E7.
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The probability of winning something on a single play at a slot machine is 0.11. After 4 plays on the slot machine, what is the probability of winning at least once
Step-by-step explanation:
The probability of winning at least once is equal to 1 minus the probability of not winning any.
P(x≥1) = 1 - P(x=0)
P(x≥1) = 1 - (1-0.11)^4
P(x≥1) = 1 - (0.89)^4
P(x≥1) = 0.373
The probability is approximately 0.373.
Answer:
37.26% probability of winning at least once
Step-by-step explanation:
For each play, there are only two possible outcomes. Either you win, or you do not win. The probability of winning on eah play is independent of other plays. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
The probability of winning something on a single play at a slot machine is 0.11.
This means that [tex]p = 0.11[/tex]
After 4 plays on the slot machine, what is the probability of winning at least once
Either you do not win any time, or you win at least once. The sum of the probabilities of these events is decimal 1. So
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
We want [tex]P(X \geq 1)[/tex]. So
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{4,0}.(0.11)^{0}.(0.89)^{4} = 0.6274[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.6274 = 0.3726[/tex]
37.26% probability of winning at least once
You want to have $600,000 when you retire in 10 years. If you can earn 8% interest compounded monthly, how much would you need to deposit now into the account to reach your retirement goal?
Answer:
$270,314.08
Step-by-step explanation:
The multiplier each month is 1+0.08/12 ≈ 1.0066667, so after 120 months, the amount is multiplied by (1.0066667)^120 ≈ 2.2196402. The amount needed is ...
$600,000/2.2196402 ≈ $270,314.08
To reach a retirement goal of $600,000 in 10 years with an 8% interest rate compounded monthly, you would need to deposit approximately $277,002.66 now.
Explanation:
In this case, we're using a formula to determine the amount needed to deposit today (P) for a future goal ($600,000) using an interest rate (r) of 8% compounded monthly for ten years. The formula to use is P = F / (1 + r/n)^(nt), where:
F is the future value of the investment ($600,000) r is the annual interest rate (8% or 0.08 as a decimal) n is the number of times that interest is compounded per unit t (12 times a year for our case) t is the time the money is invested for in years (10 years).
So, you need to plug these figures into the equation: P = 600,000 / (1 + 0.08/12)^(12*10). After doing the math, you would need to deposit around $277,002.66 now to reach your retirement goal of $600,000 in ten years given an 8% annual interest rate compounded monthly.
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In terms of x, find an expression that represents the area of the shaded region. The outer square has side lengths of (x+5) and the inner square has side lengths of (x-2), as shown.
Answer:
Area = 14x + 21 square unitsStep-by-step explanation:
The formula of an area of a square with side length a:
[tex]A=a^2[/tex]
The big square:
[tex]a=x+5[/tex]
Substitute:
[tex]A_B=(x+5)^2[/tex] use [tex](a+b)^2=a^2+2ab+b^2[/tex]
[tex]A_B=x^2+2(x)(5)+5^2=x^2+10x+25[/tex]
The small square:
[tex]a=x-2[/tex]
Substitute:
[tex]A_S=(x-2)^2[/tex] use [tex](a-b)^2=a^2-2ab+b^2[/tex]
[tex]A_S=x^2-2(x)(2)+2^2=x^2-4x+4[/tex]
The area of a shaded region:
[tex]A=A_B-A_S[/tex]
Substitute:
[tex]A=(x^2+10x+25)-(x^2-4x+4)=x^2+10x+25-x^2+4x-4[/tex]
combine like terms
[tex]A=(x^2-x^2)+(10x+4x)+(25-4)=14x+21[/tex]
if the probability of an event happening is 65% then the probability this event does not occur?
You must know that percent are ALWAYS taken out of 100. This means that 100 subtracted by 65 will give the percent that this event won't happen:
100 - 65 = 35
This event has 65% probability of happening and a 35% of NOT happening
Hope this helped!
~Just a girl in love with Shawn Mendes
The Royal Fruit Company produces two types of fruit drinks. The first type is 55% pure fruit juice, and the second type is 100% pure fruit juice. The company is attempting to produce a fruit drink that contains 70% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 240 pints of a mixture that is 70%
pure fruit juice?
Answer:
First type of fruit drinks: 160 pints
Second type of fruit drinks: 80 pints
Step-by-step explanation:
Let's call A the amount of first type of fruit drinks. 5.5% pure fruit juice
Let's call B the amount of second type of fruit drinks. 100% pure fruit juice
The resulting mixture should have 70% pure fruit juice and 240 pints.
Then we know that the total amount of mixture will be:
[tex]A + B = 240[/tex]
Then the total amount of pure fruit juice in the mixture will be:
[tex]0.55A + B = 0.7 * 240[/tex]
[tex]0.55A + B = 168[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -1 and add it to the second equation:
[tex]-A -B = -240[/tex]
[tex]-A -B = -240[/tex]
+
[tex]0.55A + B = 168[/tex]
--------------------------------------
[tex]-0.45A = -72[/tex]
[tex]A = \frac{-72}{-0.45}[/tex]
[tex]A = 160\ pints[/tex]
We substitute the value of A into one of the two equations and solve for B.
[tex]160 + B = 240[/tex]
[tex]B = 80\ pints[/tex]
To make 240 pints of a mixture that is 70% pure fruit juice, you will need 160 pints of the first type of fruit drink (55% pure fruit juice) and 80 pints of the second type of fruit drink (100% pure fruit juice).
Explanation:To solve this problem, we can set up a system of equations. Let's say x represents the number of pints of the first type of fruit drink (55% pure fruit juice) and y represents the number of pints of the second type of fruit drink (100% pure fruit juice). We know that the total number of pints of the mixture is 240, so we can write the equation x + y = 240. We also know that the desired percentage of pure fruit juice in the mixture is 70%, so we can write the equation (55% * x + 100% * y) / 240 = 70%. To solve this system of equations, we can use substitution or elimination method. Let's use substitution:
From the first equation, we can solve for x in terms of y: x = 240 - y. Substituting this into the second equation, we get ((55% * (240 - y)) + 100% * y) / 240 = 70%. Simplifying the equation, we have (0.55(240 - y) + y) / 240 = 0.70. Distributing and combining like terms, we get (132 - 0.55y + y) / 240 = 0.70. Simplifying further, we have (132 + 0.45y) / 240 = 0.70. Cross multiplying, we get 132 + 0.45y = 0.70 * 240. Simplifying, we have 132 + 0.45y = 168. Multiplying 0.45 with y, we get 0.45y = 168 - 132. Subtracting 132 from 168, we get 0.45y = 36. Dividing both sides of the equation by 0.45, we get y = 36 / 0.45. Evaluating this expression, we get y = 80. So, the number of pints of the second type of fruit drink (100% pure fruit juice) needed is 80. Substituting this value back into the first equation, we can solve for x: x + 80 = 240. Subtracting 80 from both sides of the equation, we get x = 240 - 80. Evaluating this expression, we get x = 160. Therefore, the number of pints of the first type of fruit drink (55% pure fruit juice) needed is 160.
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In a survey of 520 likely voters in a certain city, 307 said that they planned to vote to reelect the incumbent mayor. What is the probability that a surveyed voter plans to vote to reelect the mayor? Write only a number as your answer. Round to two decimal places (for example: 0.43).
Answer: 0.59
Step-by-step explanation:
Probability is a measure that quantifies the likelihood that events will occur.
Probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes .
In this case, the number of desired outcomes is 307 (surveyed voters who plan to vote to reelect the mayor), and the total number of all outcomes is 520 (total of surveyed voters) .
Then, the probability that a surveyed voter plans to vote to reelect the mayor is calculated as:
[tex]\frac{307}{520}=0.59[/tex]
The probability that a surveyed voter plans to vote to reelect the mayor is 0.59.
Explanation:To find the probability that a surveyed voter plans to vote to reelect the mayor, we divide the number of surveyed voters who plan to reelect the mayor by the total number of surveyed voters.
Given that 307 out of 520 likely voters plan to reelect the incumbent mayor, the probability is:
Probability = Number of surveyed voters who plan to reelect the mayor / Total number of surveyed voters
Probability = 307 / 520 = 0.59 (rounded to two decimal places)
You're using your meter to make voltage measurements in the circuit shown in the figure above. Your meter is connected between points A and C, and you're getting a reading of 6 V on the display. What can you conclude from this reading? A. Switch S1 is open. B. Resistors R1 and R2 have equal resistance values. C. Resistor R2 has a resistance value that's twice the value of either R1. D. Switch S1 is closed.
Answer:
A. Switch S1 is Open
Step-by-step explanation:
I attach the missing figure in the image below
Since you are getting a reading of 6V which is the maximum voltage of your circuit, you can conclude that
A. Switch S1 is Open
- If the Switch S1 was closed, we would be getting a reading of 0V. This is not the case.
- Because the switch is open, there is no current going through the circuit and therefore there is not any voltage drop across the resistors. This is why their values don't affect the reading.
If (-3, y) lles on the graph of y = 3*, then y =
The line y = 3 means that x = 0.
The point (-3, y) tells me that x = -3 when y is 3.
So, y = 3 completes the point (-3, 3).
PLEASE HELP!!!!!!!!!!!!!!!
Answer:
d. (1, 5, 2)
Step-by-step explanation:
A suitable calculator can find the reduced row-echelon form for you. Some scientific calculators and many graphing calculators have this capability, as do on-line calculator. The one below is supported by ads.