Question

The surface area of a cylinder, A, in terms of the radius of the base, r, and height, h, is given by the equation A equals 2 pi r left-parenthesis r plus h right-parenthesis.



What is the radius of the base in terms of the surface area and height of the cylinder?

Answer options with 4 options
A.
r equals StartRoot StartFraction A over 2 pi EndFraction EndRoot minus StartFraction h over 2 EndFraction

B.
r equals StartRoot StartFraction A over 2 pi EndFraction minus h EndRoot

C.
r equals StartRoot StartFraction A over 2 pi EndFraction plus StartFraction h squared over 2 EndFraction EndRoot minus StartFraction h over 2 EndFraction

D.
r equals StartRoot StartFraction A over 2 pi EndFraction plus StartFraction h squared over 4 EndFraction EndRoot minus StartFraction h over 2 EndFraction

Answer 1

Start with A = 2πr(r + h) → 2πr^2 + 2πhr − A = 0. Divide by 2π: r^2 + hr − A/(2π) = 0.

Solve by the quadratic formula:
r = [-h ± sqrt(h^2 + 4A/(2π))]/2 = [-h ± sqrt(h^2 + 2A/π)]/2.

Taking the positive root (r ≥ 0) and rewriting gives
r = sqrt(A/(2π) + h^2/4) − h/2,

which is option D.