A manufacturer has determined that the weekly profit from the sale of x items is given by the function below. It is estimated that after t days in an week, x items will have been produced. Find the rate of change of profit with respect to time at the end of 7 days.
P9x) = -x^2+600x-3000 with x=1.5t^2-2t
First I tried plugging in "t" to the x equation, then "x" into the p equation, which didn't produce the right answer. Then I tried finding the derivative of the p equation and plugging in t to x and x into the derivative, which also didn't produce the correct answer
To find the rate of change of profit with respect to time at the end of 7 days, you need to use the chain rule for differentiation. Let's go through the steps step by step:
1. Start with the given equation for x in terms of t: x = 1.5t^2 - 2t.
2. Differentiate the equation with respect to t to find dx/dt (the rate of change of x with respect to time):
dx/dt = d/dt(1.5t^2 - 2t) = 3t - 2.
3. Now we have x in terms of t and dx/dt. Substitute these values into the profit function P(x):
P(x) = -x^2 + 600x - 3000.
P(t) = -((1.5t^2 - 2t)^2) + 600(1.5t^2 - 2t) - 3000.
4. Simplify the equation to get P(t) in terms of t:
P(t) = -2.25t^4 + 6t^3 + 900t^2 - 1200t - 3000.
5. Finally, differentiate the equation P(t) with respect to t to find dP/dt (the rate of change of profit with respect to time):
dP/dt = d/dt(-2.25t^4 + 6t^3 + 900t^2 - 1200t - 3000).
6. Compute the derivative using the power rule and the sum/difference rule:
dP/dt = -9t^3 + 18t^2 + 1800t - 1200.
7. Now, to find the rate of change of profit at the end of 7 days (t = 7), substitute t = 7 into the equation for dP/dt:
dP/dt (t=7) = -9(7)^3 + 18(7)^2 + 1800(7) - 1200.
8. Compute the value to get the final answer.
Note: Make sure to double-check your calculations to ensure accuracy.
To find the rate of change of profit with respect to time at the end of 7 days, we need to calculate the derivative of the profit function with respect to time, evaluate it at t = 7, and then substitute the value of x at t = 7 in the derivative.
We are given that x = 1.5t^2 - 2t.
First, let's find dx/dt (the derivative of x with respect to t).
dx/dt = d/dt(1.5t^2 - 2t)
= 3t - 2
Now, substitute the expression for x in the profit function:
P(x) = -x^2 + 600x - 3000
= -(1.5t^2 - 2t)^2 + 600(1.5t^2 - 2t) - 3000
Expand and simplify:
P(t) = -(2.25t^4 - 6t^3 + 6t^2) + (900t^2 - 1200t) - 3000
= -2.25t^4 + 6t^3 - 6t^2 + 900t^2 - 1200t - 3000
= -2.25t^4 + 6t^3 + 894t^2 - 1200t - 3000
Now, find dP/dt (the derivative of P with respect to t).
dP/dt = d/dt(-2.25t^4 + 6t^3 + 894t^2 - 1200t - 3000)
= -9t^3 + 18t^2 + 1788t - 1200
Evaluate dP/dt at t = 7:
dP/dt at t = 7 = -9(7)^3 + 18(7)^2 + 1788(7) - 1200
= -2058 + 882 + 12516 - 1200
= 11040
Finally, substitute x at t = 7 to find the rate of change of profit with respect to time:
rate of change of profit with respect to time at t = 7 = 11040