When to use chain rule and product rule?

How do I differentiate between these two. Thanks very much!!

Answer: There were 40 adult tickets and 28 children's tickets

Step-by-step explanation:

Let x represent the number of adult tickets sold.

Let y represent the number of children's tickets sold.

An event sold $608 worth of tickets adult tickets cost $11 while children's tickets cost $6. It means that

11x + 6y = 608 - - - - - - 1

If 68 tickets were sold, it means that

x + y = 68

Substituting x = 68 - y into equation 1, it becomes

11(68 - y) + 6y = 608

748 - 11y + 6y = 608

- 11y + 6y = 608 - 748

- 5y = 140

y = 140/5 = 28

x = 68 - y = 68 - 28

x = 40

Answer: the distance it can travel on full tank is 675 miles

Step-by-step explanation:

Gas Mileage is how many miles you travel on one gallon of gasoline.

A new Honda Accord Hybrid ( part electric, part gasoline ) tests for its gas mileage by traveling 432 miles on 8 gallons of gasoline.

This means that its gas mileage is

Number of miles travelled / number of gallons of gasoline used.

Gas mileage = 432/8 = 54 miles per gallon

If a full tank containing 12 1/2 = 25/2 gallons is used, the distance that it can travel will be

Gas mileage × volume of the full tank. it becomes

54 × 25/2 = 675miles

Answer:

13

Step-by-step explanation:

5-(-8)=13

Answer:

13

Step-by-step explanation:

The number line is really helpful in this case. All you have to do is count the spaces between -8, where A is, and 5, where C is. There's 13 spaces between them, therefore the length of AC is 13.

Answer:

16+h=c

Step-by-step explanation:

Answer:

The smallest poster has dimension 25.6 cm by 32.05 cm.

Step-by-step explanation:

Let "x" and "y" be the length and the width of the poster.

The margin of a poster are 4 cm and the side margins are 5 cm.

The length of the print = x - 2(4) = x - 8

The width of the print = y - 2(5) = y - 10

The area of the print = (x- 8)(y -10)

The area of the print is given as 388 square inches.

(x-8)(y -10) = 388

From this let's find y.

y -10 = [tex]\frac{388}{(x - 8)}[/tex]

y = [tex]\frac{388}{x - 8} + 10[/tex] -------------------(1)

The area of the poster = xy

Now replace y by [tex]\frac{388}{x - 8} + 10[/tex], we get

The area of the poster = x ([tex]\frac{388}{x - 8} + 10[/tex])

= [tex]10x + \frac{388x}{x - 8}[/tex]

To minimizing the area of the poster, take the derivative.

A'(x) =  [tex]10 + 388(\frac{-8}{(x-8)^{2} } )[/tex]

A'(x) = [tex]10 - \frac{3104}{(x-8)^2}[/tex]

Now set the derivative equal to zero and find the critical point.

A'(x) = 0

[tex]10 - \frac{3104}{(x-8)^2}[/tex] = 0

[tex]10 = \frac{3104}{(x-8)^2}[/tex]

[tex](x - 8)^2 = \frac{3104}{10}[/tex]

[tex](x - 8)^2 = 310.4[/tex]

Taking square root on both sides, we get

x - 8 = 17.6

x = 17.6 + 8

x = 25.6

So, x = 25.6 cm takes the minimum.

Now let's find y.

Plug in x = 25.6 cm in equation (1)

y =  [tex]\frac{388}{25.6 - 8} + 10[/tex]

y = 22.05 + 10

y = 32.05

Therefore, the smallest poster has dimension 25.6 cm by 32.05 cm.

The minimum area of the poster given that the printed area is 388 sq.cm and the margins are 4cm top/bottom and 5cm on the sides, is found by using calculus and the area constraints. The problem can be solved by expressing width and height in terms of one another and then optimizing the area equation.

To solve the problem, we use calculus and the area constraint to find the dimensions with a minimum area. Let's denote that width of the poster is w and height is h. The entire area of the poster would then be expressed as:

(w+2*5cm)*(h+2*4cm) = poster area

On the other hand, we have a fixed constraint that states the printed area is 388cm², so we have:

w*h = 388 cm²

We can express h as 388/w and substitute this in the poster's area equation, and then use calculus to find the minimum area given these constraints.

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Check the picture below.

The probability of guessing correctly within ten tries is 10/1,000,000, simplifying to 1/100,000.

This problem is related to probability. The total number of ways to form a six-digit code with numbers from 0 to 9, where numbers can be repeated, is 10^6 because there are 10 possible choices for each of the 6 places. Thus there are 1,000,000 possible codes.

The probability of you correctly guessing the code on any one try would then be 1/1,000,000. If you try ten times, each attempt independent of the others, you still have a 1/1,000,000 chance each try. Combining these ten independent events, the total probability of guessing the correct code in ten tries would be 10/1,000,000.

Therefore, your probability is 10/1,000,000, which simplifies to 1/100,000.

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Answer:

  (B)  9/20

Step-by-step explanation:

The fastest machine can do 1/4 of the job in 1 hour. The second-fastest machine can do 1/5 of the job in 1 hour. Together, these two machines can do ...

  (1/4) +(1/5) = (5+4)/(5·4) = 9/20

of the job in 1 hour.

9/20 of the job can be done in 1 hour by two of the machines.

Answer:

a) [tex]P(\bar X>81.5)=1-0.933=0.067[/tex]

b) [tex]P(\bar X<69.5)=0.0062[/tex]

c) [tex]P(73.4<\bar X<84.05)=0.8755[/tex]  

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:

[tex]X \sim N(\mu=77,\sigma=27)[/tex]  

And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:

[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]

On this case  [tex]\bar X \sim N(77,\frac{27}{\sqrt{81}})[/tex]

Part a

We want this probability:

[tex]P(\bar X>81.5)=1-P(\bar X<81.5)[/tex]

The best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

If we apply this formula to our probability we got this:

[tex]P(\bar X >81.5)=1-P(Z<\frac{81.5-77}{\frac{27}{\sqrt{81}}})=1-P(Z<1.5)[/tex]

[tex]P(\bar X>81.5)=1-0.933=0.067[/tex]

Part b

We want this probability:

[tex]P(\bar X\leq 69.5)[/tex]

If we apply the formula for the z score to our probability we got this:

[tex]P(\bar X \leq 69.5)=P(Z\leq \frac{69.5-77}{\frac{27}{\sqrt{81}}})=P(Z<-2.5)[/tex]

[tex]P(\bar X\leq 69.5)=0.0062[/tex]

Part c

We are interested on this probability

[tex]P(73.4<\bar X<84.05)[/tex]  

If we apply the Z score formula to our probability we got this:

[tex]P(73.4<\bar X<84.05)=P(\frac{73.4-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{84.05-\mu}{\frac{\sigma}{\sqrt{n}}})[/tex]

[tex]=P(\frac{73.4-77}{\frac{27}{\sqrt{81}}}<Z<\frac{84.05-77}{\frac{27}{\sqrt{81}}})=P(-1.2<z<2.35)[/tex]

And we can find this probability on this way:

[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)[/tex]

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)=0.9906-0.1151=0.8755[/tex]

The cost per gallon of regular gasoline will be $2.69 per gallon.

The rate is the ratio of the amount of something to the unit. For example - If the speed of the car is 20 km/h it means the car travels 20 km in one hour.

MOBIL Service station sells various types of fuel: standard, besides, and premium. Mr.Adams burned through $36.25 on 12.5 gallons of standard fuel at the MOBIL Service station.

The cost per gallon for regular gasoline will be given as,

Rate = $36.25 / 12.5

Rate = $2.69 per gallon

The cost per gallon of regular gasoline will be $2.69 per gallon.

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Answer:

x = 12

Step-by-step explanation:

Given expression is \[7x-9-11=3x+4+2x\]

Simplifying: \[7x -(9+11) = (3x+2x)+4\]

Or, \[7x - 20 = 5x + 4\]

Bringing all terms containing x to the left side of the equation and all the numeric terms to the right side:

\[7x-5x = 20 + 4\]

=> \[(7-5)x = 24\]

=> \[2x = 24 \]

=> x = \[\frac{24}{2}\]

=> x= 12

Hence the value of x which satisfies the given equation \[7x-9-11=3x+4+2x\] is 12

Answer:

each paid $2.45

Step-by-step explanation:

9.80/4 = 2.45

Answer:

8.5 seconds to hit the ground

Step-by-step explanation:

A soccer ball is thrown upward from the top of a 204 foot high building at a speed of 112 feet per second.

[tex]h(t)=-16t^2+V_0t+h_0[/tex]

Vo is the speed 112 feet per second

h0 is the initial height = 204 foot

So the equation becomes

[tex]h(t)=-16t^2+112t+204[/tex]

When the soccer ball hit the ground then the height becomes 0

[tex]0=-16t^2+112t+204[/tex]

Apply quadratic formula

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

[tex]t=\frac{-112+\sqrt{112^2-4 (-16) \cdot 204}}{2(-16)}[/tex]

[tex]t=\frac{-112+\sqrt{25600}}{-32}=-1.5[/tex]

[tex]\frac{-112-\sqrt{25600}}{-32}=8.5[/tex]

time cannot be negative

so it takes 8.5 seconds to hit the ground

The statements that must be true are:

a. m∠CEA = 90°

b. m∠CEF = m∠CEA + m∠BEF

e. ∠AEF is a right angle.

Here's why:

a. m∠CEA = 90°

b. m∠CEF = m∠CEA + m∠BEF

c. m∠ CEB = 2(m∠ CEA)

d. ∠ CEF is a straight angle.

e. ∠ AEF is a right angle.

Answer:

There are 6 teams will play for each team.

Step-by-step explanation:

Given;

Total amount of money = $ 2040

Cost for blue team = $ 160

Cost for red team = $ 180

Solution,

Let number of blue teams be x and the  number of red teams be y.

Total cost =[tex]160x + 180y = 2040[/tex]

Since it is a league match and so both divisions must have equal teams.

∴ x=y

∴ [tex]160x + 180x = 2040[/tex]

[tex]340x = 2040\\x = 6[/tex]

Hence the number of teams in each division is 6.

Final answer:

Sally looked at 48 turkey trees and found 7/8 were too small. By multiplying 48 by the fraction 7/8, we find that 42 trees were too small.

Explanation:

The question asks us to calculate the number of turkey trees that were too small, based on the total number of trees Sally looked at and the fraction that were too small.

Sally looked at 48 trees in total and found that 7/8 of them were too small. To find the number of too small trees, we multiply the total number of trees by the fraction that were too small:

Number of too small trees = Total number of trees × Fraction too small

Number of too small trees = 48 × 7/8

Calculating this gives us:

Number of too small trees = 48 × 0.875

Number of too small trees = 42

Therefore, out of the 48 turkey trees Sally looked at, 42 of them were too small for her Thanksgiving needs.

Answer:

option B)[tex]\frac{1}{12}[/tex]

Step-by-step explanation:

According to the given conditions, the total number of outcomes are 24.

The probability that of drawing hearts card is

P=[tex]\frac{(total number of hearts cases)}{(total number of cases)}[/tex]

total number of hearts cases= 6

thus P= [tex]\frac{6}{24} = \frac{1}{4}[/tex]

now the probability of 4 or 6 is,

P=[tex]\frac{(total number of 4 or 6 cases)}{(total number of cases)}[/tex]

thus P= [tex]\frac{8}{24} = \frac{1}{3}[/tex]

thus, by multiplication law,

final probality is P= [tex](\frac{1}{4})(\frac{1}{3})[/tex]

P= [tex]\frac{1}{12}[/tex]

Answer:

12,000

Step-by-step explanation:

espero que he ayudado

Answer and explanation:

Given : Suppose you and a friend each choose at random an integer between 1 and 8, inclusive.

The sample space is

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)  (1,7) (1,8)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)   (2,7) (2,8)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)   (3,7) (3,8)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)   (4,7) (4,8)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)   (5,7) (5,8)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)   (6,7) (6,8)

(7,1) (7,2) (7,3) (7,4) (7,5) (7,6)   (7,7) (7,8)

(8,1) (8,2) (8,3) (8,4) (8,5) (8,6)   (8,7) (8,8)

Total number of outcome = 64

To find : The following probabilities ?

Solution :

The probability is given by,

[tex]\text{Probability}=\frac{\text{Favorable outcome }}{\text{Total outcome}}[/tex]

a) p(you pick 5 and your friend picks 8)

The favorable outcome is (5,8)= 1

[tex]\text{Probability}=\frac{1}{64}[/tex]

b) p(sum of the two numbers picked is < 4)

The favorable outcome is (1,1), (1,2), (2,1)= 3

[tex]\text{Probability}=\frac{3}{64}[/tex]

c) p(both numbers match)

The favorable outcome is (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8) = 8

[tex]\text{Probability}=\frac{8}{64}[/tex]

[tex]\text{Probability}=\frac{1}{8}[/tex]

The probabilities for choosing specific integers or pairs with a particular sum between 1 and 8 are computed based on the total number of possible combinations, which is 64 (8x8).

The probability of choosing a specific pair of integers when you and a friend each pick an integer at random between 1 and 8 can be found using the principles of probability. With 8 choices for each person, there are 64 (8 × 8) possible combinations.

To compute the probability that the sum of the two numbers picked is < 4, notice that there are very few combinations: (1,1), (1,2), (2,1). So, P(sum < 4) is 3/64.

Answer: Andre picked 252 pounds of apples

Step-by-step explanation:

Let x = number of pounds of apple picked by Jane.

Let y = number of pounds of apple picked by Andre

Let z = number of pounds of apple picked by Maria

Andre picks three times as many pounds as maria. It means that

y = 3z

Jane picks two times as many pounds as Andre. It means that

x = 2y

The total weight of the apples is 840 pounds. It means that

x + y + z = 840 - - - - - - - - - 1

We will substitute z = y/3 and x = 2y into equation 1

2y + y + y/3 = 840

Cross multiplying with 3

6y + 3y + y = 2520

10y = 2520

y = 2520/10 = 252

x = 2y = 252× 2 = 504

z = y/3 = 252/3 = 84

The intensity at a distance of 2.7 meters is 77.17 milliroentgens per hour.

Given:

I= 62.5 milliroentgens per hour

Distance = 3 meters

If the intensity of radiation varies inversely as the square of the distance from the machine, use the inverse square law formula:

[tex]I = k/d^2[/tex]

Where:

I represents the intensity of radiation,

k is a constant,

d represents the distance from the machine.

Substituting the value back to formula as

62.5 = k/(3²)

62.5 = k/9

k = 62.5 x 9

k = 562.5

So, the intensity at a distance of 2.7 meters:

I = 562.5/(2.7²)

I = 562.5/7.29

I = 77.17 milliroentgens per hour

Therefore, the intensity is 77.17 milliroentgens per hour.

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The intensity of radiation from a machine at a distance of 2.7 meters is 77.16 milliroentgens per hour, according to the inverse square law in Physics.

The question is related to inverse square law in Physics. The intensity (ℤ) of radiation varies inversely as the square of the distance (d) from the machine. Mathematically, this relationship is represented as ℤ = k/d^2 where k is a constant. Given that at a distance of 3 meters, the intensity is 62.5 milliroentgens per hour, we can find the constant k = ℤ * d^2, i.e., k = 3^2 * 62.5 = 562.5.

Now, you want to know the intensity at a distance of 2.7 meters, which we can find by substituting this value and the constant k in our equation: ℤ = k/d^2, which results in ℤ = 562.5/(2.7^2) = 77.16 milliroentgens per hour. Therefore, the intensity at a distance of 2.7 meters from the radiation machine is 77.16 milliroentgens per hour.

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Answer:

Around 100,i.e. 105

Step-by-step explanation:

In the given cube

Length of the cube = 7 unit cells

Breadth of the cube = 3 unit cells

Height of the cube = 5 unit cells

Therefore the number of unit cubes required to make such big cube is nothing but the volume of the big cube = length*breadth*height

⇒Number of cubes used to make that big cube= 7*5*3

                                                                               = 105

Hence, option D (around 100) is the correct answer

Answer:

The answer to your question is  y = x - 23

Step-by-step explanation:

Process

1.- Get the slope of the line

If two lines are parallels, it means that they have the same slope.

                                 y = 1x - 10

Slope = m = 1

2.- Get the equation of the line

                               y - y1 = m(x - x1)

                               y + 29 = 1(x + 6)                 Substitution

                               y + 29 = x + 6                     Expanding

                               y = x + 6 - 29                      Simplifying

                               y = x - 23

Answer:

y = x - 23

Step-by-step explanation:

All lines parallel to the given line y = x - 10 have the same slope (1), and the same form of equaiton:  y = x + C, where C is a constant.

We know that the new line passes through (-6, -29).  Replacing x with -6, y with -29, we get:

-29 = -6 + C, and thus we find that C = -23.

Thus, the desired equation is

y = x - 23

Answer:

  39 cups

Step-by-step explanation:

If we assume that the 3.5 cups of concentrate make 3.5+3 = 6.5 cups of tea, we can use the proportion ...

  6.5/3.5 = x/21

to find the x cups of tea Ahab can make with 21 cups of concentrate.

Multiplying by 21, we get ...

  x = 21(6.5/3.5) = 39

Ahab can make 39 cups of tea.

Answer:

Below.

Step-by-step explanation:

If g(x) is the inverse of g(x) then  g(f(x)) = x.

g(f(x)) =  4 (x + 9)/ 4 - 9

= x + 9 - 9

= x.

So it is the inverse.

Also if you find f((g(x)) it is also = to x.

Answer:

The result is [tex]-\frac{371}{6}[/tex]

Step-by-step explanation:

The given expression is:

[tex]\frac{7}{-2.5+8.5}+\frac{0.6(8+13)}{-0.2}[/tex]

First, we solve the denominator of the left-hand fraction

[tex]\frac{7}{6}+\frac{0.6(8+13)}{-0.2}[/tex]

Then we find the sum in the numerator of the right-hand fraction

[tex]\frac{7}{6}+\frac{0.6(21)}{-0.2}[/tex]

Now we solve the second fraction

[tex]\frac{7}{6}-63[/tex]

The final result is the subtraction of both expressions

[tex]-\frac{371}{6}[/tex]

Answer:

We conclude that cheddar popcorn weighed less than 5.5 ounces.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ =5.5 ounces

Sample mean, [tex]\bar{x}[/tex] = 5.23 ounces

Sample size, n = 64

Alpha, α = 0.05

Sample standard deviation, σ = 0.24 ounce.

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 5.5\text{ ounces}\\H_A: \mu < 5.5\text{ ounces}[/tex]

We use One-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{5.23 - 5.5}{\frac{0.24}{\sqrt{64}} } = -9[/tex]

Now, [tex]z_{critical} \text{ at 0.05 level of significance } =-1.645[/tex]

Since,  

[tex]z_{stat} < z_{critical}[/tex]

We reject the null hypothesis and accept the alternate hypothesis. Thus, we conclude that cheddar popcorn weighed less than 5.5 ounces.

The z test is mathematically given as z=-1.86, and a>p value.

therefore we failed to accept null hypothesis.

Question Parameters:

A random sample of 64 bags

weighed, on average, 5.23 ounces

the standard deviation of 0.24 ounce.

µ = 5.5 ounces

Generally, hypothesis  is mathematically given as

H_0:\mu = 5.8 null

H_a:\mu < 5.8 alter

Using z test

[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}\\\\z=\frac{5.74-5.8}{\frac{0.26}{\sqrt{65}}}[/tex]

z=-1.86

In conclusion,

p value = 0.0314

α = 0.10

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Answer:

The number of games sold by Chris  = [tex](\frac{g}{8}  - 17)[/tex]

Step-by-step explanation:

The total number of employees in the team = 8

The total number of games sold by the whole team = g

The number of games sold by Chris = Average Games sold by each member - 17

Now, [tex]\textrm{Average number of games sold by each}  = \frac{\textrm{Total number of games sold by team}}{\textrm{Total number of people in team}}\\[/tex]

=[tex]\frac{g}{8}[/tex]

⇒The average number of games sold by each of the team member = g/8

Hence, the number of games sold by Chris  = [tex](\frac{g}{8}  - 17)[/tex]

Answer:

The answer to your question is r = 2x

Step-by-step explanation:

Volume of a cylinder = V = π r² h

r = radius

h = height = x

Volume = 4x³

Then

                  r² = [tex]\frac{volume}{\pi h}[/tex]

                  r² = [tex]\frac{4x^{3} }{\pi x}[/tex]

                  r² = 4x²

                  r = 2x