Find the rate of change of total revenue, cost, and profit with respect to time. Assume that R9x) and C9x) are in dollars.
R(x)=60x-0.5x^2
C(x)=5x+15
x=30 and dx/dt=20 units per day
well, what is P?
That is the rate you are calculating.
Hmm. Ignore my previous comment. Still, if the functions are as given, Reiny's answer appears correct. Are there mis-types somewhere?
To find the rate of change of total revenue, cost, and profit with respect to time, we need to calculate their derivatives.
The total revenue function is given by R(x) = 60x - 0.5x^2. To find dR/dt, we can differentiate R(x) with respect to x and then multiply by dx/dt:
dR/dt = (dR/dx) * (dx/dt)
Differentiating R(x) with respect to x, we get:
dR/dx = d/dx (60x - 0.5x^2)
= 60 - x
Substituting x = 30 and dx/dt = 20 units per day, we have:
dR/dt = (60 - 30) * 20
= 30 * 20
= 600 units per day
Therefore, the rate of change of total revenue with respect to time is 600 units per day.
Similarly, the total cost function is given by C(x) = 5x + 15. To find dC/dt, we can differentiate C(x) with respect to x and then multiply by dx/dt:
dC/dt = (dC/dx) * (dx/dt)
Differentiating C(x) with respect to x, we get:
dC/dx = d/dx (5x + 15)
= 5
Substituting dx/dt = 20 units per day, we have:
dC/dt = 5 * 20
= 100 units per day
Therefore, the rate of change of total cost with respect to time is 100 units per day.
To find the rate of change of profit with respect to time, we can use the formula:
dP/dt = dR/dt - dC/dt
Substituting the values we found earlier, we have:
dP/dt = 600 - 100
= 500 units per day
Therefore, the rate of change of profit with respect to time is 500 units per day.
Correct answer 600
rate of change of total revenue is 600
rate of change of total cost is 100
rate of change of total profit is 500
Profit(x) = R(x) - C(x)
= 60x - .5x^2 - 5x - 15 = -.5x^2 + 55x - 15
dPdt (x) = -x dx/dt+ 55
= -30(20) + 55
= .....