Check my work? The equations given were R(x)=-x^2+400x and C(x)=x^2+40x+100. We were asked to find the maximum profit and minimum average cost. (I'm having trouble remembering how to find a minimum and maximum, so please help with that, if you can!)
For Max Profit, I got $16,100.
For Min Avg. Cost, I got $20, which feels really wrong.
Help?
P(x) = R(x)-C(x) = -2x^2+360x-100
P'(x) = -4x+360
P'=0 at x=90
P(90)=16100
good work
AvgCost
A(x) = C(x)/x = x+40+100/x
A'(x) = 1-100/x^2
A'=0 at x=10
A(10) = 60
Ohhhh. I see what I did wrong in that second one!
Thanks for all your help!(:
To find the maximum profit and minimum average cost, we need to analyze the given revenue function R(x) and cost function C(x).
To start, let's find the maximum profit. The profit function can be derived by subtracting the cost function from the revenue function:
P(x) = R(x) - C(x)
In this case, the profit function is:
P(x) = (-x^2 + 400x) - (x^2 + 40x + 100)
= -x^2 + 400x - x^2 - 40x - 100
= -2x^2 + 360x - 100
To find the maximum profit, we need to find the value of x that maximizes the profit function. One way to find the maximum or minimum of a quadratic function is by using the vertex formula. The vertex formula states that for a quadratic function of the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by x = -b / (2a).
In this case, the quadratic equation is -2x^2 + 360x - 100. Comparing this with the vertex formula, we can see that a = -2 and b = 360. Plugging these values into the formula, we have:
x = -b / (2a)
x = -360 / (2 * -2)
x = -360 / -4
x = 90
So the value of x that maximizes the profit function is 90. Now, substitute this value back into the profit function to find the maximum profit:
P(x) = -2x^2 + 360x - 100
P(90) = -2(90)^2 + 360(90) - 100
P(90) = -2(8100) + 32400 - 100
P(90) = -16200 + 32400 - 100
P(90) = 16100
Therefore, the maximum profit is $16,100, which matches your result.
Now let's find the minimum average cost. The average cost is given by:
AC(x) = C(x) / x
In this case, the average cost function is:
AC(x) = (x^2 + 40x + 100) / x
= x + 40 + 100/x
To find the minimum average cost, we need to find the value of x that minimizes the average cost function. One method to find the minimum or maximum of a rational function is by taking its derivative, setting it equal to zero, and solving for x.
Differentiating the average cost function, we have:
AC'(x) = 1 - 100/x^2
Setting AC'(x) equal to zero and solving for x, we have:
1 - 100/x^2 = 0
1 = 100/x^2
x^2 = 100
x = ±10
Since x represents the number of units produced, it cannot be negative. Therefore, x = 10.
Substituting this value back into the average cost function, we have:
AC(x) = x + 40 + 100/x
AC(10) = 10 + 40 + 100/10
AC(10) = 10 + 40 + 10
AC(10) = 60
Therefore, the minimum average cost is $60, which is different from your result of $20. It seems you made a mistake in your calculations for the minimum average cost.
Hence, the correct results are:
- Maximum profit: $16,100
- Minimum average cost: $60