1. ## Mathematical Models (PreCalc)

These are a few review problems I was given for a upcoming test and I'm completely stumped how to do these.

My main problem is the concept of "Expressing :blank: as a function of :blank:".

You can answer the problems completely if you'd like ( learning for my self would benefit me more but I could always do other problems ) but I just need a very dumbed down explanation.

Example 1)

The demand for a certain commodity is given $D(x) = -50x + 800;$ that is, x units of the commodity well be demanded by consumers when the price is $p = D(x)$ dollars pre unit. Total consumer expenditure $E(x)$ is the amount of money consumers pay to buy x units of the commodity.

A: Express consumer expenditure as a function of x, and sketch the graph of $E(x)$.

B: Use the in part A to determine the level of production x at which consumer expenditure is largest. What price $p$ corresponds to maximum consumer expenditure?

Example 2)

A spherical cell of radius r has volume $V = 4/3 pi r$
and surface are $S = 4 pi r^2$.

Express V as a function of S. If S is doubled, what happens to V?

Thanks for any help any one gives.

2. Originally Posted by ARR0624
These are a few review problems I was given for a upcoming test and I'm completely stumped how to do these.

My main problem is the concept of "Expressing :blank: as a function of :blank:".

You can answer the problems completely if you'd like ( learning for my self would benefit me more but I could always do other problems ) but I just need a very dumbed down explanation.

Example 1)

The demand for a certain commodity is given $D(x) = -50x + 800;$ that is, x units of the commodity well be demanded by consumers when the price is $p = D(x)$ dollars pre unit. Total consumer expenditure $E(x)$ is the amount of money consumers pay to buy x units of the commodity.

A: Express consumer expenditure as a function of x, and sketch the graph of $E(x)$.
since the price is $D(x)$, the cost for buying $x$ units will be $xD(x)$ ...do you see why?

B: Use the in part A to determine the level of production x at which consumer expenditure is largest. What price $p$ corresponds to maximum consumer expenditure?
your answer to the last question should give you the formula for a parabola. the x coordinate of the vertex of this parabola is the level of production x at which consumer expenditure is largest. the y coordinate of the vertex of this parabola is the price that corresponds to maximum consumer expenditure

Example 2)

A spherical cell of radius r has volume $V = 4/3 pi r$
and surface are $S = 4 pi r^2$.

Express V as a function of S. If S is doubled, what happens to V?

Thanks for any help any one gives.
$V = \frac 43 \pi r^3$

$S = 4 \pi r^2$

we will try to put S in V

$V = \frac 43 \pi r^3 = \frac 13 \cdot {\color {red}4 \pi r^2} \cdot r$

can you take it from here?

3. Wow, yeah that helps so much. I just had one of those "Omg Why didn't I see that" moments.

I really appreciate it.