A right circular cylinder has a height of 12\text{ centimeters}. If the total surface area… — SAT

A right circular cylinder has a height of $12\text{ centimeters}$ . If the total surface area of the cylinder is $170\pi\text{ square centimeters}$ , what is the volume, in cubic centimeters, of the cylinder?

A
$120\pi$
$300\pi$ Cevap
C
$2040\pi$
D
$3468\pi$

Cevap

The volume of the cylinder is

300\pi

cubic centimeters.

The correct answer is the volume of

300\pi

cubic centimeters. We find this by using the cylinder's surface area equation

2\pi r h + 2\pi r^2 = 170\pi

with

h = 12

, which reduces to

r^2 + 12r - 85 = 0

. Factoring this equation gives the radius

r = 5

. Plugging

r = 5

and

h = 12

into the volume formula

V = \pi r^2 h

gives

300\pi

Adım Adım Çözüm

Set up the formula for the total surface area of a cylinder and plug in the given values.

2\pi r h + 2\pi r^2 = 170\pi

, where

h = 12

. This simplifies to

2\pi r(12) + 2\pi r^2 = 170\pi

, or

24\pi r + 2\pi r^2 = 170\pi

We need to find the radius of the cylinder to calculate its volume.

Divide the entire equation by

2\pi

to simplify it into a standard quadratic form.

r^2 + 12r - 85 = 0

Simplifying the equation makes it easier to solve for the radius

r

Factor the quadratic equation to find the positive value of the radius

r

(r + 17)(r - 5) = 0

, which gives

r = 5

since the radius must be positive (

r > 0

Factoring allows us to solve for the variable

r

Use the radius

r = 5

and height

h = 12

to compute the volume of the cylinder.

V = \pi r^2 h = \pi (5^2)(12) = 25 \times 12 \pi = 300\pi

cubic centimeters.

The volume formula for a cylinder is

V = \pi r^2 h

Anahtar Kavram

Calculating volume from total surface area for a cylinder by solving a quadratic relation.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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