A division of Ditton Industries manufactures a deluxe toaster oven. The daily ma

Question

A division of Ditton Industries manufactures a deluxe toaster oven. The daily marginal profit function associated with the production and sales of the deluxe toaster ovens is known to be p'(x) = -0.0003x^2 + 0.08x + 22 Where x denoted the number of units manufactured and sold daily and p'(x) is measured in dollars per unit. Management has determined that the daily fixed daily fixed cost uncured in the production is $800. Find the total profit realizable from the manufacture and sale of 200 units of the toaster ovens per day. What is the additional daily profit realizable if the production and sale of the toaster ovens are increased from 200 to 220 units/day? The U.S. national debt stood at $10.025 trillion in 2008 (t = 0) and was growing at the rate of R(t) = 0.070251 t^2 - 0.51548t + 2.1667 trillion dollars/year in year t between 2008 and 2012. What was the U.S. national debt in 2012? (Round your answer to three decimal places.)

Explanation / Answer

Answer:

Marginal profit function is
P'(x) = - (0.0003)x^2+0.08x+22
to get total profit function Integrate on both sides
profit function is
P(x) = Integral [- (0.0003)x^2+0.08x+22].dx
P(x) = - (0.0003).(x^3/3)+(0.08).(x^2/2)+22x+C
P(x) = - (0.0001)x^3+(0.04)x^2+22x+C
when x= 0 profit P(x) = 0, then
0 = - 0+0+0+C
C = 0
profit function
P(x) = -(0.0001)x^3+(0.04)x^2+22x
when x = 200
P = -(0.0001)(200)^3+(0.04)(200)^2+22(200) = -(0.0001)(8000000)+(0.04)(40000)+4400
P = - 800+1600+4400=5200
daily fixed cost = 800
total profit = 5200-800 = $4400

b) when x = 220

P(220) = -0.0001(220)^3 + 0.04(220)^2 + 22(220)

P= $ 5711.2

C)

Given R(t)=0.070251t^2 - 0.51548t+2.1667     ( 0 <= t <=4 )

Integrate with respect to t on both sides , we get

integral of R(t) dt = 0.070251(t^3)/3 - 0.51548(t^2)/2+2.1667t = D(t)

Then U.S national debt in 2012 is for t = 4

D(4) = 0.070251(4)^3/3 - 0.51548(4^2)/2+2.1667(4)

       = 6.041648

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