pre calc question

Question

Quonset huts are temporary, all-purpose structures shaped like half-cylinders.

You have 1100 square feet of material to build a quonset hut

- The surface area S of a quonset hut is given by S = πr2 + πrl. Substitute 1100 for S and then write an expression for l in terms of r.

- The volume of V of a quonset hut is given by V = 12πr2l. Write an equation that gives V as a function in terms of r only.

-Find the value of r that maximizes the volume of the hut. Round your answer to the nearest tenth.

Answer 1

First question, solve for L in terms of R.

S = πr² + πrL

We know S = 1100, so

1100 = πr² + πrL

We want to get L by itself. Subtract πr²

1100 - πr² = πrL

Now divide by πr.

(1100 - πr²) / (πr) = L

Second question, write Volume in terms of r only

V = 12 π r² L

Replace "L" with what we solved for above: (1100 - πr²) / (πr)

V = 12 π r² (1100 - πr²) / (πr)

Not sure if they want you to simplify that or if the above equation will suffice.

Third, find the value of r that maximizes the volume of the hut. The above equation is a quadratic, as it has both an r² and r term. So the maximum volume is going to be the vertex of the parabola.

If you can use a graphing calculator, graph the V equation above and use trace to find the vertex. The x-value of the vertex refers to the radius (the y-value is the volume)

If your teacher is mean, then you'll have to simplify the equation and get it into ax² + bx + c = 0 form and solve using the quadratic equation. That will get messy!