The cost to produce x items is given by the function C(x) = 1200 + 8x + 0.01x^2.
a) Write a function that represents the average cost of producing x items.
b) Use the derivative of the function from a) to find the number of items that should be produced to minimize the average cost.
(a) A(x) = C(x)/x = 0.01x + 8 + 1200/x
(b) dA/dx = 0.01 - 1200/x^2
so find where dA/dx=0
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
You are purchasing four items and want to calculate the tax. The items cost $2.50, $8.75, $3.00, and $10.25. The tax rate is 6%. How much is the tax? Write in distribution property and without.
a) The average cost of producing x items can be calculated by dividing the total cost (C(x)) by the number of items (x).
Therefore, the function that represents the average cost is:
AC(x) = C(x) / x
Substituting the given cost function into the equation above:
AC(x) = (1200 + 8x + 0.01x^2) / x
b) To find the number of items that should be produced to minimize the average cost, we need to find the minimum point of the average cost function. This can be done by taking the derivative of the average cost function and setting it equal to zero.
AC'(x) = (d/dx) ( (1200 + 8x + 0.01x^2) / x )
= ( (d/dx) (1200 + 8x + 0.01x^2) * x - (1200 + 8x + 0.01x^2) * (d/dx) (x) ) / x^2
= ( 8 + 0.02x - (1200 + 8x + 0.01x^2) ) / x^2
= ( 8 + 0.02x - 1200 - 8x - 0.01x^2 ) / x^2
= (0.02x - 0.01x^2 - 1192) / x^2
Setting AC'(x) = 0 and solving for x:
(0.02x - 0.01x^2 - 1192) / x^2 = 0
0.02x - 0.01x^2 - 1192 = 0
0.02x - 0.01x^2 = 1192
0.01x^2 - 0.02x + 1192 = 0
Unfortunately, this is a quadratic equation that does not have a simple factored form. Therefore, to find the number of items that should be produced to minimize the average cost, we need to use numerical methods such as graphing or using a calculator to find the x-value at which the average cost function reaches its minimum value.
To find the average cost, we divide the total cost by the number of items produced.
a) The average cost function can be represented as:
AC(x) = C(x) / x
To find the average cost function, substitute the given cost function C(x) into the equation:
AC(x) = (1200 + 8x + 0.01x^2) / x
Simplifying this expression, we have:
AC(x) = (1200/x) + 8 + 0.01x
b) To find the number of items that should be produced to minimize the average cost, we can use the derivative of the average cost function, which will give us the rate of change of the average cost with respect to x.
Let's differentiate the average cost function with respect to x:
dAC(x)/dx = d/dx[(1200/x) + 8 + 0.01x]
To differentiate this, we need to differentiate each term separately:
dAC(x)/dx = d/dx(1200/x) + d/dx(8) + d/dx(0.01x)
The derivative of 1200/x is -1200/x^2, the derivative of 8 is 0, and the derivative of 0.01x is 0.01.
Substituting these derivatives back into the equation, we have:
dAC(x)/dx = -1200/x^2 + 0 + 0.01
To find the number of items that should be produced to minimize the average cost, we set the derivative equal to zero and solve for x:
-1200/x^2 + 0.01 = 0
Rearranging the equation, we have:
1200/x^2 = 0.01
Cross-multiplying:
0.01x^2 = 1200
Dividing both sides by 0.01:
x^2 = 120,000
Taking the square root of both sides:
x = sqrt(120,000)
Therefore, to minimize the average cost, the number of items that should be produced is approximately 346.41.