Question

3. An engineer is designing a cylindrical metal tank that is to hold 500 cubic feet of

gasoline.

(a) Write the height h as a function of the radius r. Hint: the volume of a cylinder
is found using V = πr²h.

(b) Use the result of part (a) to write the surface area S as a function of r, and graph
the function. Please attach your graph at the end of the test. Hint: the surface
area of a cylinder is found using S = 2πr² + 2πrh.

(c) Find the radius that minimizes the surface area (using desmos or your other
graphing calculator). Indicate the minimum point on the graph from part (b). List here the length of that radius to the nearest tenth of a foot.

(d) If the tank costs $8 per square foot of metal to construct, then what is the
minimum cost for making the tank?

Answer 1

Sure, let's work through this step by step!

(a) To write the height h as a function of the radius r, we can use the formula for the volume of a cylinder, which is V = πr²h. In this case, the volume is given as 500 cubic feet. So, we have:

500 = πr²h

To isolate h, we divide both sides of the equation by πr²:

h = 500 / (πr²)

(b) Now, let's write the surface area S as a function of r. The formula for the surface area of a cylinder is S = 2πr² + 2πrh. Using the expression for h from part (a), we can substitute it into the formula:

S = 2πr² + 2πr(500 / (πr²))

Simplifying further:

S = 2πr² + 1000/r

(c) To find the radius that minimizes the surface area, we can use a graphing calculator or software like Desmos. By graphing the function S = 2πr² + 1000/r, we can find the minimum point on the graph. The length of that radius will give us the answer.

(d) Once we have the radius that minimizes the surface area, we can substitute it back into the equation for the surface area, S = 2πr² + 1000/r, to find the minimum cost for making the tank. The cost is given as $8 per square foot of metal, so we multiply the surface area by 8 to get the minimum cost.

Let me know if you have any questions or need further assistance!