2.) An office supply company sells x mechanical pencils per year at $p per penci
ID: 2838276 • Letter: 2
Question
2.) An office supply company sells x mechanical pencils per year at $p per pencil. The price-demand equation for these pencils is p=10-.001x, and the annual cost C(x)=5000+2x.
a.) How many pencils should be produced to maximize revenue?
b.) What is the company's maximun revenue
c.) What price should the company charge for the pencils to maximize revenue?
Please show all work.
1.) For the function f(x)= x^3 - 2x^2 + x find the absolute extrema on the interval [-1,1] 2.) An office supply company sells x mechanical pencils per year at $p per pencil. The price-demand equation for these pencils is p=10-.001x, and the annual cost C(x)=5000+2x. a.) How many pencils should be produced to maximize revenue? b.) What is the company's maximun revenue c.) What price should the company charge for the pencils to maximize revenue? Please show all work.Explanation / Answer
1.
f(x) = x3 - 2x2 + x
f'(x) = 3x2 - 4x + 1
f''(x) = 6x - 4
for critical points, f'(x) = 0
3x2 - 4x + 1 = 0
3x2 - 3x - x + 1 = 0
(x - 1)(3x - 1) = 0
x = 1/3, 1
f''(1/3) = 6*1/3 - 4 = -2.............point of maxima
f''(1) = 6*1 - 4 = 2...................point of minima
Absolute maximum value = f(1/3) = (1/3)3 - 2(1/3)2 + 1/3
Absolute maximum value = 1/27 - 1/9 + 1/3
Absolute maximum value = 7/27
Absolute minimum value = f(0) = 0
2.
Revenue = Price * Quantity
Revenue = p * x
Revenue = (10 - .001x) * x
= 10x - .001x^2
Profit = Revenue - Cost
= 10x - .001x^2 - [5000 + 2x]
= 8x -.001x^2 - 5000
So we have the profit function
Profit = 8x -.001x^2 - 5000
To Maximize Profit, take the derivative and set it equal to zero.
-.002x + 8 = 0
x = 4,000 mechanical pencils ............part (a)
Now put this into the demand equation and solve for p
p = 10 - .001x [at x = 4,000]
p = 10 -.001(4000)
p = $6...........part (c)
Also put x = 4,000 into the profit equation
Profit = 8(4,000) -.001(4000)^2 - 5000
Profit = $11,000 ..........part (b)
Answer: The company should produce 4,000 pencils at $6 a piece in order to realize a maximum profit of $11,000.
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