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)= 55x - 0.5x^2 , C(x)=5x + 20, when x=40 and dx/dt=30 units per day
1. The rate of change of total revenue is $? per day.
2. Find total cost with respect to time.
3. Find total profit with respect to time.
dR/dt = (55 - x) dx/dt
dC/dt = 5 dx/dt
P=R-C, so dP/dt = (50-x) dx/dt
See what you can do with that.
To find the rate of change of the total revenue, cost, and profit with respect to time, we need to take the derivative of the revenue, cost, and profit functions with respect to time and evaluate them at the given value of x and dx/dt.
1. Rate of change of total revenue:
The total revenue function R(x) = 55x - 0.5x^2 represents the revenue generated as a function of x. To find the rate of change of total revenue with respect to time, we need to find dR/dt, which is the derivative of R(x) with respect to time.
Taking the derivative of R(x) with respect to x:
dR/dx = d/dx (55x - 0.5x^2)
= 55 - x
We know that dx/dt = 30 units per day, and we are given x = 40. So, substituting these values into the derivative, we have:
dR/dt = (dR/dx) * (dx/dt)
= (55 - 40) * 30
= 15 * 30
= 450 dollars per day
Therefore, the rate of change of total revenue is $450 per day.
2. Total cost with respect to time:
The total cost function C(x) = 5x + 20 represents the cost incurred as a function of x. To find the total cost with respect to time, we need to find dC/dt, which is the derivative of C(x) with respect to time.
Taking the derivative of C(x) with respect to x:
dC/dx = d/dx (5x + 20)
= 5
Since the total cost is constant and does not depend on time, the rate of change of the total cost with respect to time is zero.
3. Total profit with respect to time:
The total profit function P(x) is given by P(x) = R(x) - C(x). To find the total profit with respect to time, we need to find dP/dt, which is the derivative of P(x) with respect to time.
Since we have already found dR/dt and dC/dt, we can calculate dP/dt as:
dP/dt = dR/dt - dC/dt
= 450 - 0 (as dC/dt is zero)
= 450 dollars per day
Therefore, the rate of change of total profit is $450 per day.
To find the rate of change of total revenue, we need to find the derivative of R(x) with respect to time (t).
1. The rate of change of total revenue is given by dR/dt.
R(x) = 55x - 0.5x^2
Differentiating both sides with respect to t:
dR/dt = dR/dx * dx/dt
dx/dt = 30 (given)
Differentiating R(x) with respect to x:
dR/dx = 55 - x
Therefore, dR/dt = (55 - x) * (dx/dt)
Plugging in x = 40 and dx/dt = 30:
dR/dt = (55 - 40) * 30 = 450 per day.
2. The total cost with respect to time is C(x) * t, where t is the time in days.
C(x) = 5x + 20
Total cost, TC(t) = C(x) * t
Substituting x = 40 and t = 1 (since we want the cost per day):
TC(t) = (5 * 40 + 20) * 1 = 220 dollars per day.
3. To find the total profit with respect to time, we subtract the total cost from the total revenue.
Total profit, P(t) = R(x) - TC(t)
Substituting the given values:
P(t) = (55x - 0.5x^2) - (5x + 20) * t
P(t) = 55x - 0.5x^2 - (5x + 20) * t
Substituting x = 40 and t = 1:
P(t) = 55 * 40 - 0.5 * 40^2 - (5 * 40 + 20) * 1 = 1200 dollars per day.
Therefore,
1. The rate of change of total revenue is $450 per day.
2. The total cost with respect to time is $220 per day.
3. The total profit with respect to time is $1200 per day.