A fan of country music plans to make a custom CD with 14 of her 25 favorite songs. How many different combinations of 14 songs are​ possible? Is it practical to make a different CD for each possible​ combination?

Answer:

4.6

Step-by-step explanation:

Use the distance formula:

d = [tex]\sqrt{(x-x)^{2}-(y-y)^{2} }[/tex]

plug in the numbers

[tex]\sqrt{(6-1)^{2} -(5-3)^{2} }\\[/tex]

PEMDAS says do parenthisee first

[tex]\sqrt{(5)^{2} -(2)^{2} }[/tex]

now square it

[tex]\sqrt{25-4}[/tex]

subtract

[tex]\sqrt{21}[/tex]

do the calculator for this part

4.582575695

round it to get

4.6

Answer:

[tex]f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]

Step-by-step explanation:

A number is a factor of f(x) if and only if f(x) is zero for that value/number.

For the factors of a function we write the factors as x-a where a is the zero of function i.e. value at which f(x) is zero.

To write the polynomial function of minimum degree with real coefficients whose zeros include 2, -4, and 1, 3i, we find the f(x) is the product of all factors i.e x-a where a will represent the given zeros.

[tex]f(x)=(x-2)(x-(-4))(x-1)(x-3i)\\f(x)=(x-2)(x+4))(x-1)(x-3i)\\f(x)=(x^{2} +4x-2x-8)(x^{2} -3xi-x+3i )\\f(x)=(x^{2} +2x-8)(x^{2} -x-(3x+3)i)\\[/tex]

As it is stated that polynomial should have real coefficients so skipping the terms with 'i' we get

[tex]f(x)=(x^{2} +2x-8)(x^{2} -x)\\f(x)=x^{4}-x^{3}+2x^{3}-2x^{2} -8x^{2} +8x\\f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]

Answer:

f(x) = x4 - 2x2 + 36x - 80

Step-by-step explanation:

The mean of the top five test scores is 96. Then the mean absolute deviation is zero.

It is the average distance between each data point and the mean.

The top five test scores in Mr. Rhodes's class were: 98, 92, 96, 97, and 97.

Then the mean (μ) will be

[tex]\mu = \dfrac{98+92+96+97+97}{5}\\\\\mu = \dfrac{480}{5}\\\\\mu = 96[/tex]

Then the mean absolute deviation will be

[tex]\rm MAD = \dfrac{\Sigma (X_i - \mu)}{n}\\\\MAD = \dfrac{(98-96)+(92-96)+(96-96)+(97-96)+(97-96)}{5}\\\\MAD = \dfrac{2-4+0+1+1}{5}\\\\MAD = 0[/tex]

The mean absolute deviation is zero and the mean is 96.

More about the mean absolute deviation link is given below.

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Final answer:

Calculating the mean absolute deviation of the top five test scores in Mr. Rhodes's class involves finding the mean score and then the average of the absolute deviations from this mean. The result is a MAD of 1.6, indicating the average dispersion of the top scores from the class mean.

Explanation:

The question involves calculating the mean absolute deviation (MAD) of the top five test scores in Mr. Rhodes's class, which are 98, 92, 96, 97, and 97. First, we find the mean (average) of these scores by adding them together and dividing by the number of scores, which gives us (98 + 92 + 96 + 97 + 97) / 5 = 96. Next, we calculate the absolute deviation of each score from the mean, which are 2, 4, 0, 1, and 1 respectively. The mean absolute deviation is then the average of these absolute deviations, calculated as (2 + 4 + 0 + 1 + 1) / 5 = 1.6.

The result of these calculations shows that the mean absolute deviation of the test scores is 1.6. This value represents how much, on average, each of the top five scores deviates from the mean score of the class, providing an understanding of the variability or dispersion within the top scores. The MAD is a useful measure in understanding how spread out the scores are from the mean.

Answer:The answer is B. 160 + 20d

Step-by-step explanation: first find out the total for how much he makes at the first job. 20 x 8 = 160 then you just write out the equation

Answer: the distance of the boat from the base of the lighthouse is 192.9 m

Step-by-step explanation:

The scenario is represented in the right angle triangle shown in the attached photo.

Looking at triangle ABC, the height of the light house operator above sea level represents the opposite side of the right angle triangle.

Angle A = 10° because it is alternate to the angle of depression.

To determine AB, the distance of the boat from the base of the lighthouse, we would apply

the tangent trigonometric ratio which is expressed as

Tan θ, = opposite side/adjacent side. Therefore,

Tan 10 = 34/AB

AB = 34/Tan 10 = 34/0.1763

AB = 192.9 m

Answer:

2.7%.

Step-by-step explanation:.

Given:

Net profit margin ( profitability rate) =  15%

Total sales = $155 million

Total assets = $312 million

Total equity = $223 million.

Dividend rate = 10%

Question asked:

What is the firm's sustainable growth rate ?

First of all we will find these thing.

1. Asset utilization rate = [tex]\frac{Total \ sales}{Total \ assets}[/tex]

                                     = [tex]\frac{155}{312} = 0.496\ million= 0.5\%[/tex]

2. Financial utilization rate   =   [tex]\frac{Total\ debt}{Total\ equity} \\[/tex]

  Total debt = Total asset - Total equity

                   = $312 million -   $223 million = $89 million

                                           = [tex]\frac{89}{223} = 0.4\%[/tex]

3. Return on equity rate = Asset utilization rate [tex]\times[/tex] profitability rate

Return on equity rate = [tex]0.5\times15\times0.4=3\%[/tex]

4. Business retention rate = 100 - Dividend rate

                                           = 100 - 10 = 90%

Now, finally we will calculate sustainable growth rate :

Sustainable growth rate =  Return on equity rate [tex]\times[/tex]  Business retention rate

                                        = [tex]3\%\times90\%=2.7\%[/tex]

Therefore, firm's sustainable growth rate is 2.7%

Option A: [tex]y=\frac{3}{4}x+1[/tex] is the equation of the line.

Option E: [tex]y+2=\frac{3}{4}(x+4)[/tex] is the equation of the line.

Explanation:

Given that the line is [tex]3 x-4 y=7[/tex] and passes through the point [tex](-4,-2)[/tex]

We need to determine the equation of the line.

Formula:

The equation of the line can be determined using the formula,

[tex]y-y_1=m(x-x_1)[/tex]

Slope:

Since, the lines are parallel, from the equation [tex]3 x-4 y=7[/tex], we shall determine the slope.

Thus, we have,

[tex]-4 y=-3x+7[/tex]

   [tex]y=\frac{3}{4} x+\frac{7}{4}[/tex]

Thus, the slope of the equation is [tex]m=\frac{3}{4}[/tex]

Equation of line:

Substituting [tex]m=\frac{3}{4}[/tex] and the point [tex](-4,-2)[/tex] in the formula, we get,

[tex]y+2=\frac{3}{4}(x+4)[/tex]

Hence, the equation of line is [tex]y+2=\frac{3}{4}(x+4)[/tex]

Thus, Option E is the correct answer.

Let us write the equation of line [tex]y+2=\frac{3}{4}(x+4)[/tex] in slope - intercept form.

Thus, we have,

[tex]y+2=\frac{3}{4}x+3[/tex]

     [tex]y=\frac{3}{4}x+3-2[/tex]

     [tex]y=\frac{3}{4}x+1[/tex]

Thus, the equation of the line is [tex]y=\frac{3}{4}x+1[/tex]

Hence, Option A is the correct answer.

Answer:

There will be total 2048 parts of the given paper if James if able to fold the paper eleven times.

The needed function is [tex]y = 2 ^n[/tex]

Step-by-step explanation:

Let us assume the piece of paper James decides to fold is a SQUARE.

Now, let us assume:

n : the number of times the paper is folded.

y : The number of parts obtained after folds.

Now, if the paper if folded ONCE ⇒  n = 1

Also, when the pap er is folded once, the parts obtained are TWO equal parts.

⇒  for n = 1 , y = 2       ..... (1)

Similarly, if the paper if folded TWICE  ⇒  n = 2

Also, when the paper is folded twice, the parts obtained are FOUR equal parts.

⇒  for n = 2 , y = 4       ..... (2)

⇒[tex]y = 2^2 = 2^n[/tex]

Continuing the same way, if the paper is folded SEVEN times  ⇒  n = 7

So, [tex]y = 2^ n = 2^7 = 128[/tex]

⇒  There are total 128 equal parts.

Lastly,  if the paper is folded ELEVEN  times  ⇒  n = 11

So, [tex]y = 2^ n = 2^{11} = 2048[/tex]

⇒  There are total 2048 equal parts.

Hence, there will be total 2048 parts of the given paper if James if able to fold the paper eleven times.

And the needed function is [tex]y = 2 ^n[/tex]

Answer:

= 1

Step-by-step explanation:

The demand for that college will be equal to 1 or it can be said as unit elastic demand over the tuition range. This means that the demand for the college would move proportionately with the tuition range of that college, since the change in revenue is due to the change in tuition only.

Hope this helps.

Good luck and cheers.

Answer:

demand for the college is equal to 1

Step-by-step explanation:

- We know that the change in Total Revenue is only the function of change in tuition fee only.

- The change in tution fee is subjected to 15% last year multiplied by the corresponding change in demand for the college will lead to a change in total revenue.

- The relation can be expressed as:

                                         ΔTR =  Δ P *ΔD

Where,

         ΔTR : Change in Total Revenue

         Δ P : Change in tuition fee

         ΔD : Change in demand.

- For TR to remain unchanged then ΔTR = 0. Hence,

                                         ΔTR = 0 = Δ P *ΔD

- We are given a change in Δ P = 15%, so that means for ΔTR = 0, the change in demand ΔD = 0.

- ΔD = 0, also means that the elasticity of the demand curve is perfectly elastic or in other words the demand for the college is equal to 1.

Answer:

See attached picture.

Step-by-step explanation:

See attached picture.

For remaining parts resubmit question.  

Answer:

Step-by-step explanation:

it is 11423 ft3

Final answer:

The difference in volume of the weather balloon from launch to burst is calculated using the formula for the volume of a sphere, considering the change in diameters from 5 ft to 28 ft. The resulting difference is approximately 11,428.89 cubic feet.

Explanation:

The question asks for the difference in volume of a weather balloon when it burst compared to at launch. To solve this, we use the formula for the volume of a sphere, which is V = \(\frac{4}{3}\)\(\pi\)r^3, where r is the radius of the sphere. Given that the diameter at launch was 5 ft and at burst was 28 ft, the radiuses would be 2.5 ft and 14 ft, respectively.

Volume at launch: V1 = \(\frac{4}{3}\)\(\pi\)(2.5)^3 \approx 65.45 cubic feet. Volume at burst: V2 = \(\frac{4}{3}\)\(\pi\)(14)^3 \approx 11,494.34 cubic feet. The difference in volume: V2 - V1 \approx 11,494.34 - 65.45 \approx 11,428.89 cubic feet.

Therefore, the difference in volume of the balloon when it burst compared to at launch is approximately 11,428.89 cubic feet.

Answer:

F=3

Step-by-step explanation:

Due to the difficulty of visualizing the graph of the function in degrees (graph 1), we will graph it in radians (graph 2)

f(x)=−sin(3x)−1 ≡ y=−sin(3x)−1

To graph y=−sin(3x)−1

y=a.sin(bx+c)+d, where

a=-1, b=3, c=0, d=-1 and the period (T) of the function  is:

[tex]T=\frac{2\pi }{b}=\frac{2\pi }{3}[/tex]

On the graph 2 we place the original function y=sin(x) to compare

We watch that y=−sin(3x)−1 moves 1 down (-), but amplitud is the same (1)

Frequency is the number of repetitions (3x) of a function in a given interval, so

F=3

The frequency of the function f(x) = -sin(3x) - 1 is 3/(2π), determined by the coefficient of x inside the sine function.

The frequency of the function f(x) = -sin(3x) - 1 can be determined by examining the coefficient of x within the sine function. The standard form for a sine function is f(x) = sin(Bx), and the frequency f is given by f = B/(2π). In this case, the coefficient B is 3, so the frequency of the function is 3/(2π), which is already in simplest fraction form.

*100% CORRECT ANSWERS

Question 1

Alan is writing out the steps using the "shortest Route Algorithm". On the second step, he just circled the route ABD as the shortest route from A to D. What should he cross out next?  

AD; 6  

Question 2

Beth is writing out the steps using the "Shortest Route Algorithm". She just finished writing out all the routes for the third step. What route should she circle next?  

ACE; 6  

Question 3

If you start at vertex A and use the "shortest route" algorithm, what would be the second path to be selected/highlighted?  

ACF  

(SEE ATTACHMENTS BELOW)

Answer: ACF

Step-by-step explanation: Starting at vertex A and using shortest routes algorithm the secondary route to be selected would be ACF = 1+2 = 3

The first route would be AC = 1. A vertex is a point where two straight lines meet or join, they are usually found in angles.

The quadratic function in vertex form to model the height of the flea compared to the horizontal distance travelled is h = -2/15*(d-15)² + 30 with the maximum point at (15, 30) and passing through the origin.

The problem here can be diagnosed using concepts of

quadratic functions

and

vertex form

. In a real world scenario, the motion of a projectile like the flea jumping can be modeled using a downward opening parabola represented by a quadratic function. In this case, we are asked to find the quadratic function in vertex form, which is given by

h = a(d - h1)² + k

where (h1,k) is the vertex of the parabola. In the given scenario, the maximum height attained by the flea is 30 cm which is at a horizontal distance of 15 cm from the starting point, thus the vertex of the parabola is (15, 30). From the information given, we know that the flea starts from the ground, so at the origin, height h = 0. Substituting these values, we get the equation of the parabola as

h = -a(d-15)^2 + 30

.

To find the value of 'a'

, we can use the information that the parabola passes through the origin (0,0). Substituting these values in the equation, we get a = -30/225 = -2/15. Therefore, the quadratic function in vertex form to model the height of the flea compared to the horizontal distance travelled becomes

h = -2/15*(d-15)² + 30

.

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Answer:

Correct option is (a) equal to the population mean.

Step-by-step explanation:

According to the Central limit theorem if a large sample is selected from an unknown population with mean μ and standard deviation σ then the sampling distribution of sample means follows a normal distribution.

The mean and standard deviation of this sampling distribution is:

[tex]\mu_{\bar x}=\mu[/tex]

[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}[/tex]

This standard deviation of the sampling distribution of mean is known as the standard error.

Thus, the correct option is (a) equal to the population mean.

The mean of the sampling distribution of the sample mean is equal to the population mean, as stated by the Central Limit Theorem, when the sample size is sufficiently large.

The correct answer to the student's question is: a. equal to the population mean. This principle is a key concept in statistics, known as the Central Limit Theorem. According to the theorem, if the size (n) of the sample is sufficiently large, the distribution of the sample means will be approximately normal, with the mean of these sample means equalling the population mean. Another important point to note is that the standard deviation of this distribution, called the standard error of the mean, is the population standard deviation divided by the square root of the sample size (n). Therefore, as the sample size increases, the standard deviation of the sampling distribution of the means decreases, leading to more precise estimations of the population mean.

The slope of Line m that is parallel to Line l is also 4/9.

The slope of a line determines its steepness and direction. When two lines are parallel, they have the same slope. In this case, Line l is parallel to Line m, and the slope of Line l is 4/9. Therefore, the slope of Line m is also 4/9.

Step-by-step explanation:

ok so your gonna need to set up a proportion, specifically

25 is to x as x is to 16

mathematically, it's a cross multiplying problem:

25                X

           =          

X                   16

cross multiply and you get 400 = x²

which when solved is 20.

Have a good night dude.

Answer:

[tex]2\frac{1}{6}[/tex] pounds.

Step-by-step explanation:

We have been given that Gina's literacy bucket weighs 6 pounds. Her novel weighs 1 4/6 pounds, and her chrome book weighs 2 1/6.

To find weight of bucket after taking out the two items, we will subtract weight of each item from 6 pounds as:

[tex]\text{Weight of bucket}=6-1\frac{4}{6}-2\frac{1}{6}[/tex]

Let us convert mixed fractions into improper fractions as:

[tex]1\frac{4}{6}=\frac{6\cdot 1+4}{6}=\frac{10}{6}\\\\2\frac{1}{6}=\frac{6\cdot 2+1}{6}=\frac{12+1}{6}=\frac{13}{6}[/tex]

[tex]\text{Weight of bucket}=6-\frac{10}{6}-\frac{13}{6}[/tex]

[tex]\text{Weight of bucket}=\frac{6\cdot 6}{6}-\frac{10}{6}-\frac{13}{6}[/tex]

[tex]\text{Weight of bucket}=\frac{36}{6}-\frac{10}{6}-\frac{13}{6}[/tex]

Combine numerators:

[tex]\text{Weight of bucket}=\frac{36-10-13}{6}[/tex]

[tex]\text{Weight of bucket}=\frac{13}{6}[/tex]

[tex]\text{Weight of bucket}=2\frac{1}{6}[/tex]

Therefore, the weight of the bucket is [tex]2\frac{1}{6}[/tex] pounds.

Answer:

After 12 years height of both the trees would be same.

Step-by-step explanation:

Given,

Height of tree type A = 5 ft

Height of tree type B = 8 ft

We need to find after how many years both the trees will be of same height.

Solution,

Firstly we will convert the height of both plants into inches.

Since we know that 1 feet is equal to 12 inches.

So height of tree type A =[tex]5\ ft=5\times12=60\ in[/tex]

Similarly, height of tree type B =[tex]8\ ft=8\times12=96\ in[/tex]

Also given that;

Rate of growth of tree type A = 9 in/year

and rate of growth of tree type A = 6 in/year

Let the number of years be 'x'.

So according to question after 'x' years the height of both trees type A and type B will be same.

Now we can frame the equation as;

[tex]60+9x=96+6x[/tex]

Combining the like terms, we get;

[tex]9x-6x=96-60\\\\3x=36[/tex]

On dividing both side by '3' using division property, we get;

[tex]\frac{3x}{3}=\frac{36}{3}\\\\x=12[/tex]

Hence after 12 years height of both the trees would be same.

Answer:

y=2x+7

Step-by-step explanation:

2x−y+4=0

y=2x+4

m=2

Line are parallel, so their slope is the same.

A(-1,5)... x1 =-1,y1 =5

y-y1 =m(x-x1)

y1 - 5=2(x-(-1))

y1 - 5=2(x+1)

y1-5=2x+2

y=2x+2+5

y=2x+7

Answer: y = 2x + 7

Step-by-step explanation:

The equation of a straight line can be represented in the slope intercept form as

y = mx + c

Where

c represents y intercept

m represents the slope of the line.

The equation of the given line is

2x - y + 4 = 0

y = 2x + 4

Comparing with the slope intercept form, slope = 2

If two lines are parallel, it means that they have the same slope. Therefore, the slope of the line passing through (- 1, 5) is 2

To determine the y intercept, we would substitute m = 2, x = - 1 and y = 5 into y = mx + c. It becomes

5 = 2 × - 1 + c

5 = - 2 + c

c = 5 + 2 = 7

The equation becomes

y = 2x + 7

Answer:

a) For Bella, x has to be a positive even values

For Tito, x has to be a negative even values

b) For Bella, x = 4

For Tito, x = -4

Answer:

B.

Step-by-step explanation:

The line is going up 1 and over 2, making the slope 1/2 and the y-intercept is 1. Hope this helped!

Answer:

C

Step-by-step explanation:

The x-axis is more than 7.0 and the y-axis is less than 7.0.

Cheers

The well is 35 feet deep.

Step-by-step explanation:

Given,

Length of rope = 50 feet

Length of remaining rope = 15 feet

Let,

x be the depth of well.

Total length of rope = Length of remaining + Depth of well

[tex]50=15+x\\50-15=x\\x=35[/tex]

The well is 35 feet deep.

Step-by-step explanation:

Let us assume the two needed vehicles are A and Q.

Let P(A) be the probability of the vehicle A available when needed.

And, P(Q) be the probability of the vehicle Q available when needed.

Now, P(A) = 90 % = 0.90

⇒ P (not A) =  1 - P(A)  

                    =  1- 0.9 =  0.1

⇒ P (not A) = 0.1

Similarly,  P(Q) = 90 % = 0.90

⇒ P (not Q) =  1 - P(Q)  

                    =  1- 0.9 =  0.1

⇒ P (not Q) = 0.1

So, the probability that both the vehicles are NOT available when needed  

= P(not A) x P(not Q)  

= 0.1 x 0.1 = 0.01

Hence, the probability that neither vehicle is available at a given time is 0.01

The probability of an eighth-grade student using the Internet, based on the data provided, is approximately 0.62, or simply 8/13 as a fraction.

To calculate the probability that a student uses the Internet given he or she is in eighth grade, we look at the eight graders' behaviors in the table. There are 13 students in total in the eighth grade, and 8 of these use the Internet.

So, the probability can be expressed as P(Internet|8th grader) = number of internet users in 8th grade / total number of 8th graders = 8 / 13.

When rounded to two decimal places, this probability would be approximately 0.62. In terms of fractions, it can simply be left as 8/13.

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The probability that a student uses the Internet, given that he or she is in eighth grade, is 2/3.

We are asked for the probability that a student uses the Internet, given that he or she is in eighth grade. This is the conditional probability [tex]$P({\text{Internet}} | {\text{8th Grade}})$[/tex], which can be calculated using Bayes' Theorem: \begin{align*}

[tex]P({\text{Internet}} | {\text{8th Grade}}) &= \frac{P({\text{8th Grade}} | {\text{Internet}})P({\text{Internet}})} {P({\text{8th Grade}})} \\&= \frac{\frac{8}{28} \cdot \frac{14}{28}} {\frac{13}{28}} \\&= \frac{2}{3}[/tex]

\end{align*}

Therefore, the probability that a student uses the Internet, given that he or she is in eighth grade, is [tex]$\boxed{\frac{2}{3}}$[/tex].

Here is a more detailed explanation of how we arrived at our answer:

Step 1: Calculate the probability of being in eighth grade, given that the student uses the Internet. This is the conditional probability [tex]$P({\text{8th Grade}} | {\text{Internet}})$[/tex], which can be calculated from the table as follows: [tex]$P({\text{8th Grade}} | {\text{Internet}}) = \frac{8}{14}$[/tex]

Step 2: Calculate the probability of using the Internet.** This is the marginal probability [tex]$P({\text{Internet}})$[/tex], which can be calculated from the table as follows: [tex]$P({\text{Internet}}) = \frac{14}{28}$[/tex].

Step 3: Calculate the probability of being in eighth grade.This is the marginal probability [tex]$P({\text{8th Grade}})$[/tex], which can be calculated from the table as follows: [tex]$P({\text{8th Grade}}) = \frac{13}{28}$[/tex].

Step 4: Apply Bayes' Theorem to calculate the conditional probability [tex]$P({\text{Internet}} | {\text{8th Grade}})$[/tex].

Therefore, the probability that a student uses the Internet, given that he or she is in eighth grade, is[tex]$\boxed{\frac{2}{3}}$[/tex].

For such more questions on probability

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Answer:

  see below

Step-by-step explanation:

The rotation transformations are ...

  90° : (x, y) ⇒ (-y, x)

  180° : (x, y) ⇒ (-x, -y)

  270° : (x, y) ⇒ (y, -x)

Applying these to the given points, you get ...

9) A'(6, 9)

10) A'(15, 11)

11) A'(9, 6)

12) A'(-11, 15)

13) A'(6, -9)

14) A'(15, -11)

Answer:

area = 8a^2 +4ab

Step-by-step explanation:

Area = (4a)(2a + b)

= 8a^2 +4ab

Step-by-step explanation:

[tex]Area of rectangle \\ = base \times height \\ = (4a) \times (2a + b) \\ = 4a \times 2a + 4a \times b \\ = 8 {a}^{2} + 4ab \\ [/tex]

Answer:

  see below

Step-by-step explanation:

In the attachment, the points are listed in the order given in the problem statement. (They are listed to the right of the "rotation matrix", with x-coordinates above y-coordinates.)

__

I really don't like to do repetitive calculations, so I try to use a graphing calculator or spreadsheet whenever possible. Angles are measured CCW.

As always, the rotation transformations are ...

  180° — (x, y) ⇒ (-x, -y)

  270° — (x, y) ⇒ (y, -x)

Answer:

It'll take the robot 8 minutes to complete one task

7.5 tasks will be completed in one hour

Step-by-step explanation:

Total time to complete 5 tasks is 2/3hr (40 minutes)

Time it takes to complete one task = 40 ÷ 5 = 8 minutes

Since the robot completes one task in 8 minutes, x tasks will be completed in 60 minutes.

x = 60 ÷ 8 =  7.5 tasks

Answer:

A. 8 minutes; B. 7.5 tasks in one hour; C. It takes the robot about [tex] \\ \frac{2}{15}\;hour[/tex] or about 0.1333 hour to complete one task or 13.33% of one hour.

Step-by-step explanation:

Part A

Two thirds hours is

[tex] \\ \frac{2}{3}*60 = 40\;min[/tex]  

We know that each task takes the same amount of time. So, 40min can be divided by 5:

[tex] \\ \frac{40}{5} = 8\;min[/tex]

Thus, each task takes 8 min to be completed. Then, it takes the robot 8 minutes to complete one task.

Part B

The robot can complete 5 tasks in 40 minutes, how many tasks can the robot complete in 60 minutes or one hour?

There are 20 minutes ahead to complete one hour. In the next 8 minutes, the robot can complete one task. There are still 12 minutes ahead. In the next 8 minutes, the robot completes another task. There is still 4 minutes ahead to complete the hour, but in 4 minutes the robot can complete half of the task because it takes 8 minutes for a complete task. Therefore, the robot can complete 5 tasks + 2 tasks + 0.5 task = 7.5 tasks in one hour or 60 minutes.

We can obtain the same answer using proportions. That is, if 5 tasks are completed in 40 minutes, how many of them will be completed in one hour or 60 minutes.  

Then

[tex] \\ \frac{5\;tasks}{40\;min} = \frac{x}{60\;min}[/tex]

[tex] \\ \frac{5\;tasks}{40\;min}*60\;min = x[/tex]

[tex] \\ x = \frac{5\;tasks*60\;min}{40\;min}[/tex]

[tex] \\ x = \frac{300\;tasks}{40} = 7.5\;tasks[/tex]

Part C (A)

From part A, we already know that the robot can complete a task in 8 minutes, which is a fraction of one hour. What is this fraction? In one hour we have 60 minutes, then

[tex] \\ 8\;min*\frac{1\;hour}{60\;min} = 1\;hour*\frac{8}{60} = 1\;hour*\frac{4}{30} = 1\;hour*\frac{2}{15} = 0.1333333....\;hours \approx 0.1333\;hours[/tex].

Therefore, it takes the robot about [tex] \\ \frac{2}{15}\;hour[/tex] or 0.1333 hour to complete one task (rounding to four decimal places) or 13.33% of one hour.