suppose that for a company manufacturing calculators, the cost and revenue equations (in dollars) are given…

suppose that for a company manufacturing calculators, the cost and revenue equations (in dollars) are given by $c = 80000 + 20x$, $r = 400x-\frac{x^{2}}{3000}$, where the production output in one week is $x$ calculators. if the production rate is increasing at a rate of 500 calculators per week when the production output is 4000 calculators, find each of the following with correct units. round your answer to two decimal places if necessary. (note: profit is revenue minus costs.) the rate of change of the cost with respect to time is the rate of change of the revenue with respect to time is the rate of change of the profit with respect to time is

suppose that for a company manufacturing calculators, the cost and revenue equations (in dollars) are given by $c = 80000 + 20x$, $r = 400x-\frac{x^{2}}{3000}$, where the production output in one week is $x$ calculators. if the production rate is increasing at a rate of 500 calculators per week when the production output is 4000 calculators, find each of the following with correct units. round your answer to two decimal places if necessary. (note: profit is revenue minus costs.) the rate of change of the cost with respect to time is the rate of change of the revenue with respect to time is the rate of change of the profit with respect to time is

Answer

Explanation:

Step1: Differentiate cost function with respect to (x)

The cost function is (C = 80000+20x). Using the power - rule (\frac{d}{dx}(ax + b)=a) (where (a = 20) and (b = 80000)), we have (\frac{dC}{dx}=20). By the chain - rule (\frac{dC}{dt}=\frac{dC}{dx}\cdot\frac{dx}{dt}). Given (\frac{dx}{dt}=500), then (\frac{dC}{dt}=20\times500 = 10000) dollars per week.

Step2: Differentiate revenue function with respect to (x)

The revenue function is (R = 400x-\frac{x^{2}}{3000}). Using the power - rule (\frac{d}{dx}(ax^{n})=nax^{n - 1}), we get (\frac{dR}{dx}=400-\frac{2x}{3000}=400-\frac{x}{1500}). When (x = 4000), (\frac{dR}{dx}=400-\frac{4000}{1500}=400-\frac{8}{3}=\frac{1200 - 8}{3}=\frac{1192}{3}). By the chain - rule (\frac{dR}{dt}=\frac{dR}{dx}\cdot\frac{dx}{dt}). Substituting (\frac{dR}{dx}=\frac{1192}{3}) and (\frac{dx}{dt}=500), we have (\frac{dR}{dt}=\frac{1192}{3}\times500=\frac{596000}{3}\approx198666.67) dollars per week.

Step3: Recall profit formula and differentiate with respect to (t)

Since (P=R - C), then (\frac{dP}{dt}=\frac{dR}{dt}-\frac{dC}{dt}). We know (\frac{dC}{dt}=10000) and (\frac{dR}{dt}\approx198666.67). So (\frac{dP}{dt}=198666.67 - 10000=188666.67) dollars per week.

Answer:

The rate of change of the cost with respect to time is (10000) dollars per week. The rate of change of the revenue with respect to time is approximately (198666.67) dollars per week. The rate of change of the profit with respect to time is approximately (188666.67) dollars per week.