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
Well, it seems like you're having trouble finding the rate of change of profit with respect to time. Let's see if I can help you out, with a touch of humor!
To find the rate of change of profit with respect to time, we need to take the derivative of the profit function. So, let's tackle it step by step.
First, we need to find dx/dt. You already have the equation x = 1.5t^2 - 2t. To find the derivative, we use the power rule and the constant multiple rule. So, dx/dt = 3t - 2. Make sure to double-check your math there, I'm tricky but not that tricky!
Now, let's find dP/dx. The profit function is P(x) = -x^2 + 600x - 3000. Taking the derivative, we get dP/dx = -2x + 600. Remember, don't drop the negative sign like a clumsy clown!
Finally, to find the rate of change of profit with respect to time, we need to multiply dP/dx by dx/dt. So, we have:
dP/dt = (dP/dx) * (dx/dt)
dP/dt = (-2x + 600) * (3t - 2).
Now, plug x = 1.5t^2 - 2t into the equation and simplify to find the rate of change of profit with respect to time at the end of 7 days.
But remember, math is serious business, so don't clown around too much while solving it!
To find the rate of change of profit with respect to time, we need to find dP/dt, the derivative of the profit function P(x) with respect to time.
Given that x = 1.5t^2 - 2t, we can substitute this expression for x in the profit function:
P(x) = -x^2 + 600x - 3000
P(x) = -(1.5t^2 - 2t)^2 + 600(1.5t^2 - 2t) - 3000
Now, we can simplify this equation by expanding and collecting like terms:
P(x) = -2.25t^4 + 6t^3 - 600t^2 + 1200t - 3000
Next, we need to find the derivative of P(x) with respect to time (t):
dP/dt = d/dt (-2.25t^4 + 6t^3 - 600t^2 + 1200t - 3000)
Now, let's find the derivative of each term individually:
dP/dt = -9t^3 + 18t^2 - 1200t + 1200
Finally, to find the rate of change of profit with respect to time after 7 days (t = 7), substitute t = 7 into the derivative:
dP/dt = -9(7)^3 + 18(7)^2 - 1200(7) + 1200
Calculate the expression above to find the rate of change of profit with respect to time after 7 days.
To find the rate of change of profit with respect to time at the end of 7 days, you need to calculate the derivative of the profit function P(x) with respect to t. Let's go step-by-step:
1. Given that x = 1.5t^2 - 2t, substitute this expression for x 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
2. Simplify the expression:
P(t) = -2.25t^4 + 6t^3 - 600t^2 + 1200t - 3000
3. Now, differentiate P(t) with respect to t to find the rate of change of profit:
dP/dt = -9t^3 + 18t^2 - 1200t + 1200
4. To find the rate of change of profit at the end of 7 days, substitute t = 7 into the derivative:
dP/dt at t = 7 = -9(7)^3 + 18(7)^2 - 1200(7) + 1200
Calculate this expression, and you will find the rate of change of profit with respect to time at the end of 7 days.