Assume that it costs Apple approximately
C(x) = 36,100 + 100x + 0.01x^2
dollars to manufacture x 30-gigabyte video iPods in a day.
How many iPods should be manufactured in order to minimize average cost?
What is the resulting average cost of an iPod? (Give your answer to the nearest dollar.)
cost per unit = cpu = C(x) / x
= 36,100/x + 100 +.01 x
d cpu/dx = 0 at min
= -36,100/x^2 + .01
.01 x^2 = 36,100
x^2 = 3,610,000
x = 1900 units
now go back and get cpu
= 36,100/1900 + 100 + .01(1900)
= 2019
d cpu/dx = 0 at min
= -36,100/x^2 + .01
.01 x^2 = 36,100
x^2 = 3,610,000
x = 1900 units
now go back and get cpu
= 36,100/1900 + 100 + .01(1900)
What is the resulting average cost of an iPod?
138
To find the number of iPods that should be manufactured in order to minimize the average cost, we need to determine the derivative of the cost function and set it equal to zero. Let's start by finding the derivative of the cost function C(x):
C(x) = 36,100 + 100x + 0.01x^2
To find the derivative, we will differentiate each term with respect to x. The derivative of a constant (36,100) is zero, and we can use the power rule to differentiate the other terms:
dC(x)/dx = d(36,100)/dx + d(100x)/dx + d(0.01x^2)/dx
dC(x)/dx = 0 + 100 + 0.02x
Now, we set the derivative equal to zero and solve for x:
0.02x + 100 = 0
0.02x = -100
x = -100 / 0.02
x = -5000
Since it doesn't make sense to produce a negative number of iPods, we can conclude that the minimum average cost occurs when approximately 5,000 iPods are manufactured.
To find the resulting average cost, we substitute the value of x = 5000 into the cost function:
C(5000) = 36,100 + 100(5000) + 0.01(5000)^2
C(5000) = 36,100 + 500,000 + 0.01(25,000,000)
C(5000) = 36,100 + 500,000 + 250,000
C(5000) = 786,100
Therefore, the resulting average cost of an iPod is approximately $786,100.
To find the number of iPods that should be manufactured to minimize the average cost, we need to find the derivative of the cost function and set it equal to zero.
First, let's find the derivative of the cost function C(x).
C'(x) = 100 + 0.02x
Setting C'(x) = 0, we have:
100 + 0.02x = 0
Rearranging the equation, we get:
0.02x = -100
Dividing both sides by 0.02, we find:
x = -100 / 0.02
x = -5000
Since the number of iPods manufactured cannot be negative, we discard the solution x = -5000.
Now, we need to find the second derivative of the cost function to determine whether the solution is a minimum or a maximum.
C''(x) = 0.02
Since the second derivative is positive, it means that the solution is a minimum.
Therefore, to minimize the average cost, the optimal number of iPods that should be manufactured is 0 (approximately).
To find the average cost, substitute the optimal value of x into the cost function.
C(0) = 36,100 + 100(0) + 0.01(0^2)
C(0) = 36,100
The resulting average cost of an iPod is approximately $36,100.