Question

The surface area of a given cone is 1,885.7143 square inches. What is the slang height?

Answer 1

If
`r >> h`, the slang height of the cone is approximately 23.521 inches.

Explanation:

The surface area of a cone (A) is given by this formula:

`A = \pi \cdot r^(2) + 2\pi\cdot s`

Where:

`r` - Base radius of the cone, measured in inches.

`s` - Slant height, measured in inches.

In addition, the slant height is calculated by means of the Pythagorean Theorem:

`s = \sqrt{r^(2)+h^(2)}`

Where
`h` is the altitude of the cone, measured in inches. If
`r >> h`, then:

`s \approx r`

And:

`A = \pi\cdot r^(2) +2\pi\cdot r`

Given that
`A = 1885.7143\,in^(2)`, the following second-order polynomial is obtained:

`\pi \cdot r^(2) + 2\pi \cdot r -1885.7143\,in^(2) = 0`

Roots can be found by the Quadratic Formula:

`r_(1,2) = \frac{-2\pi \pm \sqrt{4\pi^(2)-4\pi\cdot (-1885.7143)}}{2\pi}`

`r_(1,2) \approx -1\,in \pm 24.521\,in`

`r_(1) \approx 23.521\,in \,\wedge\,r_(2)\approx -25.521\,in`

As radius is a positive unit, the first root is the only solution that is physically reasonable. Hence, the slang height of the cone is approximately 23.521 inches.