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) = 4x, C(x) = 0.01x^2 + 0.6x + 30, when x = 20 and dx/dt = 9 units per day The rate of change of total revenue is 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) = 4x, C(x) = 0.01x^2 + 0.6x + 30, when x = 20 and dx/dt = 9 units per day

The rate of change of total revenue is per day.

Transcript text: Part 1 of 3 HW Score: 61.9\%, 130 Points: 0 of 1 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. $R(x)=4 x, \quad C(x)=0.01 x^{2}+0.6 x+30$, when $x=20$ and $\frac{d x}{d t}=9$ units per day The rate of change of total revenue is $\$$ $\square$ per day.

Solution

Solution Steps

To find the rate of change of total revenue with respect to time, we need to use the chain rule from calculus. The total revenue function is given as $ R(x) = 4x $. We are given $\frac{dx}{dt} = 9$ units per day. The rate of change of revenue with respect to time is $\frac{dR}{dt} = \frac{dR}{dx} \cdot \frac{dx}{dt}$.

Step 1: Given Functions and Values

We are given the revenue function $ R(x) = 4x $ and the cost function $ C(x) = 0.01x^2 + 0.6x + 30 $. We need to evaluate the rate of change of total revenue when $ x = 20 $ and $ \frac{dx}{dt} = 9 $ units per day.

Step 2: Calculate the Derivative of Revenue

The derivative of the revenue function with respect to $ x $ is: \[ \frac{dR}{dx} = 4 \]

Step 3: Apply the Chain Rule

Using the chain rule, the rate of change of revenue with respect to time is given by: \[ \frac{dR}{dt} = \frac{dR}{dx} \cdot \frac{dx}{dt} \] Substituting the known values: \[ \frac{dR}{dt} = 4 \cdot 9 = 36 \]