Geometry question?

Question

1. The total area of a cone is 90 pi. The slant height is 13. Find the radius and volume of the cone. 

2. The total area of a cylinder is 168 pi and the height is 8. Find the radius of the cylinder. 

I think we're supposed to solve with quadratics but I'm not quite sure how to do that? Thanksss :)

Answer 1

1).

The total area of a cone (A) = πrl + πr²

Total area of a cone (A) = 90π.

Slant height (l) = 13.

Substitute the values of A = 90π and l = 13 in A = πrl + πr²

90π = π(r(13) + r² )

90 = 13r + r²

r² + 13r - 90 = 0.

r² + 13r - 90 = 0, is a quadratic equation, use quadratic formula to find roots of the related quadratic equation.

The solution x = [ - b ±  √ (b² - 4ac) ]/2a.

Compare the given equation with standard form of the quadratic equation ax² + bx + c = 0.

a = 1, b = 13, and c = - 90.

Substitute the values a = 1, b = 13, and c = - 90 in x = [ - b ±  √ (b² - 4ac) ]/2a.

r = [ - 13 ±  √ (13² - 4*1*(-90)) ]/2 * 1

r = [ - 13 ±  √ (169 + 360) ]/2

r = [- 13 ±  √529]/2

r = [- 13 ±  23]/2

r = [- 13 +  23]/2 and [- 13 -  23]/2

r = 10/2 = 5 and r = - 36/2 = - 18.

Therefore, radius of the cone is 5 units.

Volume of the cone (V) = 1/3[ πr ²√( l²- r² ) ].

Substritute the values π = 3.14, l = 13, and r = 5 in V = 1/3[ πr ²√( l²- r² ) ].

V = 1/3[ 3.14 * 5 ²√( 13²- 5² ) ]

= 1/3[ 3.14 * 25√(169 - 25) ]

= 1/3[ 3.14 * 25√144 ]

= 1/3[ 942 ]

= 314.

Therefore, volume of a cone is 314 cubic units.

Answer 2

2).

Total area of a cylinder (A) = 2πrh + 2πr².

Total area of a cylinder (A) = 168π.

Height (h) = 8.

Substitute the values of A = 168π and h = 8 in A = 2πrh + 2πr².

168π = 2π(8r + r²)

84 = 8r + r²

r² + 8r - 84 = 0

By factor by grouping.

r² + 14r - 6r - 84 = 0

r(r + 14) - 6(r + 14) = 0

Factor  : (r + 14)(r - 6) = 0

Apply zero product property.

r  + 14 = 0 and  r - 6 = 0

r = - 14 and r = 6

Therefore, radius of the cylinder is 6 units.