A solid metal sphere with radius r is melted down and recast into a right circular cylinde… — SAT

A solid metal sphere with radius $r$ is melted down and recast into a right circular cylinder with height $\frac{3}{4}r$ . If the total surface area of the cylinder is $k\pi r^2$ , what is the value of $k$ ?

A
$\frac{17}{9}$
B
$\frac{7}{2}$
C
$\frac{34}{9}$
$\frac{50}{9}$ Answer

Answer

\frac{50}{9}

The volume of the sphere with radius

r

V = \frac{4}{3}\pi r^3

. When it is melted and recast into a right circular cylinder with radius

R

and height

h = \frac{3}{4}r

, the volume remains the same. The volume of the cylinder is

V = \pi R^2 h

. Equating the two volumes gives

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right)

. Dividing both sides by

\pi r

yields

\frac{4}{3} r^2 = \frac{3}{4} R^2

, which simplifies to

R^2 = \frac{16}{9} r^2

. Taking the square root of both sides gives

R = \frac{4}{3}r

. The total surface area of the cylinder is given by the formula

S = 2\pi R^2 + 2\pi R h

. Substituting the values of

R

and

h

in terms of

r

yields

S = 2\pi \left(\frac{4}{3}r\right)^2 + 2\pi \left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = 2\pi \left(\frac{16}{9}r^2\right) + 2\pi r^2 = \frac{32}{9}\pi r^2 + \frac{18}{9}\pi r^2 = \frac{50}{9}\pi r^2

. Since the surface area is

k\pi r^2

, the value of

k

\frac{50}{9}

Step-by-Step Solution

Express the volume of the sphere in terms of its radius

r

V_{\text{sphere}} = \frac{4}{3}\pi r^3

This is the standard formula for the volume of a sphere.

Equate the volume of the sphere to the volume of the cylinder to find the cylinder's radius

R

in terms of

r

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right) \implies R^2 = \frac{16}{9}r^2 \implies R = \frac{4}{3}r

The volume of a cylinder is

V_{\text{cylinder}} = \pi R^2 h

. Since the sphere is melted and recast into the cylinder, their volumes must be equal. Solving for

R^2

and taking the square root gives the radius

R

Use the total surface area formula for the cylinder to express the surface area in terms of

r

A = 2\pi R^2 + 2\pi R h = 2\pi\left(\frac{16}{9}r^2\right) + 2\pi\left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = \frac{32}{9}\pi r^2 + 2\pi r^2 = \frac{50}{9}\pi r^2

The total surface area of a right circular cylinder consists of the area of the two circular bases and the lateral surface area.

Compare the calculated surface area to the given expression

k\pi r^2

to find the value of

k

k = \frac{50}{9}

Equating

\frac{50}{9}\pi r^2

and

k\pi r^2

yields the constant value

k

Key Concept

Equating volumes of solids and calculating total surface area of a cylinder

Estimated Time:2m 30s

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