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Math  /  Calculus

QuestionSuppose that for a company manufacturing calculators, the cost, and revenue equations are given by C=70000+40x,R=200−x240C=70000+40 x, \quad R=200-\frac{x^{2}}{40} where the production output in one week is xx calculators. If the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators, find each of the following:
Rate of change in cost == □\square
Rate of change in revenue == □\square
Rate of change in profit == □\square

Studdy Solution

STEP 1

1. The cost function C C is given by C=70000+40x C = 70000 + 40x .
2. The revenue function R R is given by R=200x−x240 R = 200x - \frac{x^2}{40} .
3. The production output x x is increasing at a rate of dxdt=500 \frac{dx}{dt} = 500 calculators per week.
4. The production output at the time of interest is x=6000 x = 6000 calculators.

STEP 2

1. Find the rate of change in cost.
2. Find the rate of change in revenue.
3. Find the rate of change in profit.

STEP 3

Find the rate of change in cost.
Differentiate the cost function C C with respect to time t t :
dCdt=ddt(70000+40x) \frac{dC}{dt} = \frac{d}{dt}(70000 + 40x)
Since 70000 70000 is a constant, its derivative is zero. Therefore:
dCdt=40dxdt \frac{dC}{dt} = 40 \frac{dx}{dt}
Substitute dxdt=500 \frac{dx}{dt} = 500 :
dCdt=40×500=20000 \frac{dC}{dt} = 40 \times 500 = 20000

STEP 4

Find the rate of change in revenue.
Differentiate the revenue function R R with respect to time t t :
dRdt=ddt(200x−x240) \frac{dR}{dt} = \frac{d}{dt}\left(200x - \frac{x^2}{40}\right)
Apply the derivative:
dRdt=200dxdt−140⋅2xdxdt \frac{dR}{dt} = 200 \frac{dx}{dt} - \frac{1}{40} \cdot 2x \frac{dx}{dt}
Simplify:
dRdt=200dxdt−x20dxdt \frac{dR}{dt} = 200 \frac{dx}{dt} - \frac{x}{20} \frac{dx}{dt}
Substitute x=6000 x = 6000 and dxdt=500 \frac{dx}{dt} = 500 :
dRdt=200×500−600020×500 \frac{dR}{dt} = 200 \times 500 - \frac{6000}{20} \times 500
dRdt=100000−150000 \frac{dR}{dt} = 100000 - 150000
dRdt=−50000 \frac{dR}{dt} = -50000

STEP 5

Find the rate of change in profit.
Profit P P is given by P=R−C P = R - C .
Differentiate with respect to time t t :
dPdt=dRdt−dCdt \frac{dP}{dt} = \frac{dR}{dt} - \frac{dC}{dt}
Substitute the values found:
dPdt=−50000−20000 \frac{dP}{dt} = -50000 - 20000
dPdt=−70000 \frac{dP}{dt} = -70000
The rate of change in cost is 20000 \boxed{20000} .
The rate of change in revenue is −50000 \boxed{-50000} .
The rate of change in profit is −70000 \boxed{-70000} .

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