### Instruction: A steel cone needs to be constructed that has a base radius of 8 inches and a slant height of 15 inches. What is the height of the cone that will minimize the amount of steel needed to construct it, assuming that steel cost $2.50 per square inch? Round your answer to two decimal places.### Response: To minimize the amount of steel needed, we need to minimize the surface area of the cone. The surface area of a cone is given by the formula:

Surface Area = πr(r + l)

where r is the base radius, and l is the slant height.

In this case, r = 8 inches and l = 15 inches. We also know that the height (h), radius (r), and slant height (l) are related by the Pythagorean theorem:

l^2 = r^2 + h^2

We can solve for h in terms of r and l:

h^2 = l^2 - r^2
h = sqrt(l^2 - r^2)

Now, we can substitute this expression for h into the surface area formula:

Surface Area = πr(r + sqrt(l^2 - r^2))

We are given that l = 15 and r = 8, so we can plug these values into the formula:

Surface Area = π(8)(8 + sqrt(15^2 - 8^2))

Now, we can simplify and calculate the surface area:

Surface Area = 8π(8 + sqrt(225 - 64))
Surface Area = 8π(8 + sqrt(161))
Surface Area ≈ 8π(8 + 12.69)
Surface Area ≈ 8π(20.69)
Surface Area ≈ 520.67 square inches

Now, we need to find the cost of the steel needed to construct the cone. We are given that steel costs $2.50 per square inch:

Cost = Surface Area × Price per square inch
Cost = 520.67 × $2.50
Cost ≈ $1301.68

Now, we need to find the height of the cone that will minimize the amount of steel needed. We can use the expression for h that we found earlier:

h = sqrt(l^2 - r^2)
h = sqrt(15^2 - 8^2)
h = sqrt(225 - 64)
h = sqrt(161)
h ≈ 12.69 inches

So, the height of the cone that will minimize the amount of steel needed is approximately 12.69 inches.