Question

Pls help

A right circular cone has a lateral surface area of 188.50 square inches. If its slant height lis 10 inches, then what is the volume of the cone in cubic inches?

Answer 1

Check the picture below.

notice, the slant-height is simply the hypotenuse from the pythagorean theorem.

`188.5=\stackrel{\textit{lateral area}}{\pi r√(r^2+h^2)}\implies 188.5=\pi r10\implies \cfrac{188.5}{10\pi }=r\implies \boxed{\cfrac{18.85}{\pi }=r} \\\\[-0.35em] ~\dotfill\\\\ h^2=(slant-height)^2-r^2\implies h=√((slant-height)^2-r^2) \\\\\\ h=\sqrt{10^2-\cfrac{18.85^2}{\pi^2}}\implies h=\sqrt{\cfrac{100\pi^2-18.85^2}{\pi^2}}\implies \boxed{h=\cfrac{√(100\pi^2-18.85^2)}{\pi }} \\\\[-0.35em] ~\dotfill`

`V=\cfrac{\pi }{3}\left( \cfrac{18.85}{\pi } \right)^2\left( \cfrac{√(100\pi^2-18.85^2)}{\pi } \right)\implies h\approx 301.6031084547~in^3`

Answer 2

169.04 in² (nearest hundredth)

Explanation:

Surface area of a cone =
`\pi`r² +
`\pi`r
`l`

(where r = radius of the base and
`l` = slant height)

Given slant height
`l` = 10 and surface area = 188.5

Surface area =
`\pi`r² +
`\pi`r
`l`

188.5 =
`\pi`r² + 10
`\pi`r

`\pi`r² + 10
`\pi`r - 188.5 = 0

r =
`(-10\pi +√((10\pi )^2-(4*\pi *-188.5)) )/(2\pi )` = 4.219621117...

Volume of a cone = (1/3)
`\pi`r²h

(where r = radius of the base and h = height)

We need to find an expression for h in terms of
`l` using Pythagoras' Theorem a² + b² = c², where a = radius, b = height and c = slant height

r² + h² =
`l`²

h² =
`l`² - r²

h = √(
`l`² - r²)

Therefore, substituting found expression for h:

volume of a cone = (1/3)
`\pi`r²√(
`l`² - r²)

Given slant height
`l` = 10 and r = 4.219621117...

volume = 169.0431969... = 169.04 in² (nearest hundredth)