What is the optimal number of items to produce for maximum profit in business?

[Shakattack12]

Homework Statement

A manufacturer makes a batch of n items with the cost (in dollars) of each item being:
n²-6n+35.
The manufacturer sells the items for $50 each. How many items should be produced in each batch to maximise profit.

Homework Equations

Cost = C(n) = n²-6n+35
Revenue = R(n) = 50n

The Attempt at a Solution

Profit = 50n - (n²-6n+35)
= 56n - n²-35
dP/dn = 56 - 2n
let dP/dn = 0
0 = 56 - 2n
n = 28
After testing the nature I found it to be a maximum which is all good but the textbook says the answer is 5. Where did I go wrong?

 

[Ray Vickson]

Homework Statement

A manufacturer makes a batch of n items with the cost (in dollars) of each item being:
n²-6n+35.
The manufacturer sells the items for $50 each. How many items should be produced in each batch to maximise profit.

Homework Equations

Cost = C(n) = n²-6n+35
Revenue = R(n) = 50n

The Attempt at a Solution

Profit = 50n - (n²-6n+35)
= 56n - n²-35
dP/dn = 56 - 2n
let dP/dn = 0
0 = 56 - 2n
n = 28
After testing the nature I found it to be a maximum which is all good but the textbook says the answer is 5. Where did I go wrong?

Your $p = n^2 -6n + 35$ is the cost per item, so the cost of a batch of $n$ items is $C(n) = n p$.

 

[Shakattack12]

Thank you so much! Can't believe I missed that.