# Thread: another calculus question (functions)

1. ## another calculus question (functions)

costs associated with manufacturing x products is described by:
C(x)=x^3-7x^2+18x+110
revenue for this level of manufacturing is : V(x)=3x^3-21x^2+84x+180

Profit = V(x)-C(x)
Calculate
1. expression for profit associated with manufacturing x items
2. how many items should be manufactured to minimise production costs
3. how many items need to be manufactured to maximise the profit

2. Hello, izaiyoi!

This is very straight foward . . . Exactly where is your difficulty?

Costs associated with manufacturing x products is described by:
. . $\displaystyle C(x)\:=\:x^3-7x^2+18x+110$

Revenue for this level of manufacturing is: .$\displaystyle V(x)\:=\:3x^3-21x^2+84x+180$

$\displaystyle \text{Profit }\:= \:P(x)\:=\:V(x)-C(x)$

Calculate:
1. Expression for profit associated with manufacturing $\displaystyle x$ items

They told us that: .$\displaystyle P(x) \:=\:V(x)-C(x)$ . . . didn't they?

$\displaystyle \text{So: }\;P(x) \;=\;\underbrace{\bigg[3x^3-21x^2+84x+180\bigg]}_{v(x)} - \underbrace{\bigg[x^3-7x^2+18x + 110\bigg]}_{C(X)} \;=\;\; ? ?$

2. How many items should be manufactured to minimise production costs?

We want to minimize $\displaystyle C(x).$

Solve: .$\displaystyle C'(x) \:=\:0$

3. How many items need to be manufactured to maximise the profit?

We want to maximize $\displaystyle P(x).$

Solve: .$\displaystyle P'(x) \:=\:0$