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a manufacturer produces cell phones. she estimates that by selling them for x dollars each, she will be able to sell 240−2x each week. step 2 of 2 : determine what the maximum revenue will be.

User MatteKarla
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1 Answer

11 votes
11 votes

Answer:

$7200

Step-by-step explanation:

Let y be the number of phones sold, and x the price per phone. The revenue, R, would be the product of phones sold and their price:

R = y*x

We know that y = 240-2x.

The total revenue would be:

x*y or x*(240-2x)

Revenue(R) = 240x - 2x^2

We want to find the maximum of this function. This can be done with one of two methods: graphing and Differentiating. Both are described:

Graphing

Plot R = 240x - 2x^2 and look for the vertex. See the attached plot. The vertex is at (60, 7200), which means that a price of 60/phone would result in the maximum revenue of $7200.

Differentiate

The second approach is to differentiate the equation and set it equal to 0. Differentiation allows one to determine the slope of the function at any point x. When this function reaches its maximum, the slope goes from positive (going up) to negative (going back down). At the very top the slope is 0 for a value of x. Solving the differentiated equation for x will reveal the maximum point:

R = 240x - 2x^2

dR/dx = 240 - 4x

0 = 240 - 4x [slope of 0 at the maximum]

4x = 240

x = $60/phone, as before

From y = 240−2x we can find the total units sold at $60/phone

y = 240 - 2(60)

y = 120 phones sold

Total revenue at maximum would therefore be:

R = x*y

R = ($60/phone)*(120 phones) = $7200

a manufacturer produces cell phones. she estimates that by selling them for x dollars-example-1
User Chakri Reddy
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