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
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.