Question

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)=45x-0.5x^2, C(x)=4x+15, when x=40 and dx/dt=30 units per day

Answer 1

To find the rate of change of total revenue, cost, and profit with respect to time, we first need to calculate the values of R(x), C(x), and P(x) when x = 40.

R(40) = 45(40) - 0.5(40)^2
R(40) = 1800 - 800
R(40) = $1000

C(40) = 4(40) + 15
C(40) = 160 + 15
C(40) = $175

Now, we can calculate the profit function P(x) as:
P(x) = R(x) - C(x)
P(x) = (45x - 0.5x^2) - (4x + 15)
P(x) = 45x - 0.5x^2 - 4x - 15
P(x) = 41x - 0.5x^2 - 15

Now, we can calculate the rate of change for each function with respect to time:

dR/dt = d(45x - 0.5x^2)/dt
dR/dt = 45 - x
dR/dt = 45 - 40
dR/dt = 5 units per day

dC/dt = d(4x + 15)/dt
dC/dt = 4
dC/dt = $4 per unit

dP/dt = d(41x - 0.5x^2 - 15)/dt
dP/dt = 41 - x
dP/dt = 41 - 40
dP/dt = 1 unit per day

So, the rate of change of total revenue with respect to time is $5 per day, the rate of change of total cost with respect to time is $4 per unit, and the rate of change of total profit with respect to time is $1 per day.