Please help me answer this question

Question

The cylinder can be made from steel. The top will be open and the thickness of the steel will be 0.5ft thick, making up the walls and the bottom. Steel can be purchased for $0.29 per kilogram and has a density of 7.95g/cm³. What is the cost of the steel to the nearest dollar?

Btw the surface area of the cylinder is 3456π ft² and the volume is 26973π ft³

Answer 1

Hi Mindy,

Let me walk you through the process of this problem. But let me also tell you that based on the area and volume given, there is no real solution.

The first thing you would need to do is determine the of the cylinder. The cylinder is open at the top, and ignoring the 0.5 ft lip that would go around the top of the cylinder, the area is the sum of the area of the base plus the area of the side. The base area is the area of a circle (πr²), and the area of the side is the circumference times the height (2πh). The volume is the area of the base times the height (πr²h). We can solve for h in terms of r in the volume equation, then substitute that answer into the area equation to get an equation with just one variable. One variable makes life easy, sort of.

So, solving for h in terms of r, using the volume equation, we have:

V = hπr², then divide both sides by πr² to isolate h:

h = V/πr² , now substitute that into the area equation:

A = πr² + 2πh substitute for h:

A = πr² + 2πrV/ πr² , divide the second term by πr to simplify:

A = πr² + 2V/ r

Now subtract A from both sides and multiply the entire equation by r to remove r from the denominator of the second term:

0 = πr³ + 2V - Ar, plug in 3456 for A and 26973 for V, then rearrange for standard form:

0 = πr³ - 3456r + 53946

Now we have a cubic equation with 0 as the coefficient of the squared term. We can use a cubic equation solver to determine r. Find one online, nobody memorizes the formula.

The problem is that when this equation is solved for its roots, there are two complex solutions (numbers with an imaginary component, 19 +/- 7.31i in this case), and one negative solution (-39.217). Even though there is one real number solution to the equation, it is negative and length cannot be negative. So the is no actual solution to the problem.

However, if there were a positive solution, then the next step would be to use that number as the radius, and plug it into the volume equation to solve for h. That would be:

h = V/r² , now you can determine the area of the base, and multiply that by 0.5 ft to get the volume of the base. The volume of the side is a bit more complicated. You need to find the volume of a cylinder of height h -0.5 (to take into account the thickness of the base) and subtract from that the volume of a cylinder of the same height, but with a radius that is 0.5 ft less. That would represent the empty space in side the cylinder. Once you have calculated those values, add them together to get the total volume of steel you would need.

You have a price for steel based on mass, so you need to convert the volume of steel into mass using the density of steel, 7.95 g/cm³, but your volume is in cubic feet, and you need to convert that into the metric system. One foot is equal to 30.48 cm, and cubing that gives 28,317 cm³ per ft³. So multiply you cylinder volume in cubic feet by 28,317 to get the volume in cm³. Next multiply the cm3 volume of the cylinder by 7.95 to get the mass of the cylinder in gram, and divide that by 1000 to convert to the number of kilograms of steel you need. Finally, multiply the number of kilograms of steel by its price of $0.29/kg and you have the cost of the steel you need.

Hope that helps.