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

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

500 = πr²h

To solve for h, we can rearrange the formula:

h = 500 / (πr²)

(b) Using the result from part (a), we can write the surface area S as a function of r. The formula for the surface area of a cylinder is S = 2πr² + 2πrh. Substituting the expression for h that we found in part (a), we get:

S = 2πr² + 2πr(500 / (πr²))

Simplifying further:

S = 2πr² + 1000 / r

To graph this function, you can use a graphing calculator or a graphing software.

(c) To find the radius that minimizes the surface area, you can use a graphing calculator or software to analyze the graph from part (b). Look for the lowest point on the graph. The length of that radius will give you the answer, rounded to the nearest tenth of a foot.

(d) To find the minimum cost for making the tank, we need to multiply the minimum surface area by the cost per square foot, which is $8. Once you find the radius that minimizes the surface area, you can plug it into the surface area function from part (b) and multiply it by $8 to get the minimum cost.

I hope this helps! Let me know if you have any further questions.