Question

The revenue (in dollars) from the sale of x car seats for infants is given by the following function. R(x) = 32x - 0.010x^2 0 ? x ? 3200 (A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats. (B) Use the four-step process to find R'(x). (C) Find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and interpret the results. (A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats. (Round to one decimal place as needed.) (B) R'(x) = (C) R(1000) = R'(1000) = Interpret these results. Choose the correct answer below. A. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is increasing at a rate of R'(1000) dollars per seat. B. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is decreasing at a rate of R'(1000) dollars per seat. C. This means that at a production level of 1,000 car seats, the revenue is R'(1000) dollars and is increasing at a rate of R(1000) dollars per seat.

          The revenue (in dollars) from the sale of x car seats for infants is given by the following function.
R(x) = 32x - 0.010x^2 0 ? x ? 3200
(A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(B) Use the four-step process to find R'(x).
(C) Find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and interpret the results.
(A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(Round to one decimal place as needed.)
(B) R'(x) =
(C) R(1000) =
R'(1000) =
Interpret these results. Choose the correct answer below.
A. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is increasing at a rate of R'(1000) dollars per seat.
B. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is decreasing at a rate of R'(1000) dollars per seat.
C. This means that at a production level of 1,000 car seats, the revenue is R'(1000) dollars and is increasing at a rate of R(1000) dollars per seat.

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The revenue (in dollars) from the sale of x car seats for infants is given by the following function.
R(x) = 32x - 0.010x^2 0 ? x ? 3200
(A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(B) Use the four-step process to find R'(x).
(C) Find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and interpret the results.
(A) Find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(Round to one decimal place as needed.)
(B) R'(x) =
(C) R(1000) =
R'(1000) =
Interpret these results. Choose the correct answer below.
A. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is increasing at a rate of R'(1000) dollars per seat.
B. This means that at a production level of 1,000 car seats, the revenue is R(1000) dollars and is decreasing at a rate of R'(1000) dollars per seat.
C. This means that at a production level of 1,000 car seats, the revenue is R'(1000) dollars and is increasing at a rate of R(1000) dollars per seat.

Added by Michael M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Eleni Katirtzoglou

07/10/2022

Step 1

** Given: \[ R(x) = 32x + 0.010x^2 \] Average change in revenue: \[ \frac{R(1050) - R(1000)}{1050 - 1000} \] Plugging in the values: \[ \frac{32(1050) - 0.010(1050)^2 - (32(1000) + 0.010(1000)^2)}{50} \] Simplify: \[ \frac{32(1050) - 0.01(1050)^2 - 32(1000) - Show more…

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Transcript

00:01 Hi, here we have the revenue in dollars from a sale of x car seats for infants that is given by this function over here.

00:11 R of x equals 32x minus 0 .010x square for x between 0 and 3 ,200.

00:22 First, we want to find the average change in revenue.

00:26 If production is changed from 1 ,000 car seats to 1 ,050.

00:33 So for part a, this is the answer for part a.

00:38 We have this.

00:39 We have to find this.

00:41 We have to compute this quantity.

00:43 Are the revenue at 150 minus the revenue at 1 ,000 over 1 ,050, over 1 ,050 minus 1 ,000.

00:55 So this will give us, let me just write it here, 32 times 150 minus 0 .0 points, what was that? 01 times 150 to the square and then minus 32 times 32 times 1 ,000, and then minus 32 times 1 ,000, plus 0 .01 times 1000 to the square and all these divided by 50 okay and then if we just simplify a little bit this is going to be equal 32 times 150 minus 1000 minus 0 .01 times open brackets 150 to the square minus 1 ,000 over 50 so you're going to have here 32 times 50 minus 0 .01 and then we have 1 ,0 .01 and then we have 1 ,050 1 ,000 times 150 plus 1000 over 50 so this is 32 times 30 times 50 minus 0 .01 and then in the first bracket we have 50 in the second bracket we have 2 ,050 over 50 you see here that 50 is a common factor so in and we have 32 minus 0 oops we have 32 minus 0 .01 times 2050 of course this 50 cancels out okay so 50 is a common factor at the numerator and the denominator so it cancels out and then at this very last step now we can use our calculator and we are going to see 32 minus 0 .01 times 2050 this is equal to 11 .5 so that's it this is the change in the revenue this is the answer in part a part b it says use the four step process to find our prime of x i don't really know what does it mean here a four step process this is r of x is equal here we have r of x is equal what was r of x the second is equal to 32 x minus 0 .010 x square so r prime of x is 32 minus 2 times 0 .01 x so r prime of x is 32 minus 2 times 0 .01 x so r prime of x is 32 minus 0 .02 x and this is the answer in part and then it says find the revenue and the estate rate of change of revenue at the production level 1 ,000 cards seats and interpret your results.

04:36 So now we are going to find r of x, r at 1 ,000.

04:42 R at 1 ,000 is 32 times 1 ,000.

04:48 Excuse me, 1 ,000.

04:56 So we have r at 1 ,000 is 32 times...

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