Questions: Find the rate of change of total revenue, cost, and profit with respect to time. Assume that R(x) and C(x) are in dollars. R(x) = 40x - 0.5x^2, C(x) = 3x + 15, when x = 25 and dx/dt = 20 units per day

Find the rate of change of total revenue, cost, and profit with respect to time. Assume that R(x) and C(x) are in dollars.
R(x) = 40x - 0.5x^2, C(x) = 3x + 15, when x = 25 and dx/dt = 20 units per day

Transcript text: Find the rate of change of total revenue, cost, and profit with respect to time. Assume that $R(x)$ and $C(x)$ are in dollars. $R(x)=40 x-0.5 x^{2}, \quad C(x)=3 x+15$, when $x=25$ and $d x / d t=20$ units per day

Solution

Solution Steps

Step 1: Differentiate the Revenue and Cost Functions

Given the revenue function $ R(x) = 40x - 0.5x^2 $ and the cost function $ C(x) = 3x + 15 $, we first find their derivatives with respect to $ x $:

\[ \frac{dR}{dx} = 40 - x \]

\[ \frac{dC}{dx} = 3 \]

Step 2: Calculate the Rate of Change with Respect to Time

We are given that $ \frac{dx}{dt} = 20 $ units per day and $ x = 25 $. We use these values to find the rate of change of revenue and cost with respect to time:

\[ \frac{dR}{dt} = \left(40 - 25\right) \times 20 = 300 \]

\[ \frac{dC}{dt} = 3 \times 20 = 60 \]

Step 3: Calculate the Rate of Change of Profit

The profit $ P(x) $ is given by $ P(x) = R(x) - C(x) $. Therefore, the rate of change of profit with respect to time is:

\[ \frac{dP}{dt} = \frac{dR}{dt} - \frac{dC}{dt} = 300 - 60 = 240 \]