Question

An apartment complex has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartment units is given by the following function. P(x) = -10x^2 + 1,700x - 41,000 To maximize the monthly rental profit, how many units should be rented out? _ units What is the maximum monthly profit (in dollars) realizable? $ The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation p = -0.00043x + 8 (0 ≤ x ≤ 12,000) where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by C(x) = 600 + 2x - 0.00003x^2 (0 ≤ x ≤ 20,000). To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profit is P(x) = R(x) - C(x). (Round your answer to the nearest whole number.) ___ discs/month

          An apartment complex has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartment units is given by the following function.

P(x) = -10x^2 + 1,700x - 41,000

To maximize the monthly rental profit, how many units should be rented out?
___ units

What is the maximum monthly profit (in dollars) realizable?
$____

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation

p = -0.00043x + 8    (0 ≤ x ≤ 12,000)

where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by

C(x) = 600 + 2x - 0.00003x^2    (0 ≤ x ≤ 20,000).

To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profit is P(x) = R(x) - C(x). (Round your answer to the nearest whole number.)
_____ discs/month

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Added by Matthew M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Elizabeth Waters

03/31/2022

Step 1

The derivative is P'(x) = -20x + 1,700. Show more…

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An apartment complex has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out x apartment units is given by the following function. P(x) = -10x^2 + 1,700x - 41,000 To maximize the monthly rental profit, how many units should be rented out? ___ units What is the maximum monthly profit (in dollars) realizable? $____ The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation p = -0.00043x + 8 (0 ≤ x ≤ 12,000) where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by C(x) = 600 + 2x - 0.00003x^2 (0 ≤ x ≤ 20,000). To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is R(x) = px, and the profit is P(x) = R(x) - C(x). (Round your answer to the nearest whole number.) _____ discs/month

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Transcript

00:01 So we have an apartment complex whose profit is found by the p of x function, and we want to maximize profit.

00:06 So to maximize profit, we're going to find the derivative and then set that derivative equal to zero.

00:11 So p prime of x equals negative 20, x plus 1 ,700, set it equal to zero.

00:21 And i'm setting it equal to zero because if i were to consider the graph of this function, it's going to be an inverted parabolas, something like this.

00:28 And when we find the derivative, that's the slope of the tangent line.

00:31 At a specific point, and we want that derivative to be equal to zero because that is when our maximum would occur.

00:40 So when we have a flat line on top, we have a slope of zero.

00:43 That's when our maximum is going to occur.

00:45 So now i'm going to subtract 1 ,700 from both sides.

00:49 And when i divide by negative 20, i get x equals 85.

00:53 And so my profit is maximized when i have 85 of my apartment units rented out.

00:59 And if i go back and i plug in 85 for x, in my original profit function, i can figure out how much profit that would be.

01:06 So 85 squared is going to give me 7 ,225 times negative 10.

01:13 It's going to give me negative 72 ,250 plus 1 ,700 times 85 is going to give me 144 ,500.

01:25 And then we're going to subtract 41 ,000.

01:27 So when i do that arithmetic there, i get a total profit of $31 ,250.

01:37 So my profit is maximized at 85 units, and i get a total of $31 ,250.

01:46 The second question we're asked about is if we had some compact disks, and there's a couple of equations that they give us, they give us a price equation of negative 0 .000 ,000.

01:58 43x plus 8.

01:59 So p represents the price of each disc and d x represents the demand of those disks.

02:05 So our revenue can be found by taking the price of each disc, this right here, and multiplying it by the demand or however money we're going to sell.

02:13 So p times x.

02:14 So if i were to take my price function and multiply it times x, i'm going to get my revenue.

02:20 So it's going to give me negative 0 .0043x squared plus 8x...

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