A rectangular prism has a length of 8 centimeters, a width of 6 centimeters, and a height … — SAT

A rectangular prism has a length of $8$ centimeters, a width of $6$ centimeters, and a height of $h$ centimeters. A right circular cylinder has a base radius of $3$ centimeters and a height of $h$ centimeters. If the volume of the rectangular prism is $V$ cubic centimeters and the volume of the cylinder is $W$ cubic centimeters, which of the following expressions represents the ratio of $V$ to $W$ ?

$\frac{16}{3\pi}$ Cevap
B
$\frac{16}{3}$
C
$\frac{3\pi}{16}$
D
$\frac{16}{\pi}$

Cevap

The ratio of the volume of the rectangular prism to the volume of the cylinder is

\frac{16}{3\pi}

The volume of the rectangular prism is the product of its dimensions:

V = 8 \times 6 \times h = 48h

. The volume of the right circular cylinder is calculated using the formula

\pi r^2 h

W = \pi \times 3^2 \times h = 9\pi h

. The ratio of

V

W

is represented by the fraction

\frac{V}{W} = \frac{48h}{9\pi h}

. Simplifying the fraction by canceling the common variable

h

and dividing the numerator and denominator by

3

yields the expression

\frac{16}{3\pi}

Adım Adım Çözüm

Write the expression for the volume of the rectangular prism,

V

, in terms of

h

V = 8 \times 6 \times h = 48h

The volume of a rectangular prism is calculated using the formula

\text{Volume} = lwh

, where

l

is the length,

w

is the width, and

h

is the height.

Write the expression for the volume of the cylinder,

W

, in terms of

h

W = \pi \times 3^2 \times h = 9\pi h

The volume of a right circular cylinder is calculated using the formula

\text{Volume} = \pi r^2 h

, where

r

is the base radius and

h

is the height.

Express the ratio of

V

W

as a fraction and simplify it.

\frac{V}{W} = \frac{48h}{9\pi h} = \frac{48}{9\pi} = \frac{16}{3\pi}

The ratio of

V

W

is given by

\frac{V}{W}

. The common height variable

h

cancels out from both the numerator and the denominator, and the fraction

\frac{48}{9}

simplifies to

\frac{16}{3}

by dividing both numbers by their greatest common divisor,

3

Anahtar Kavram

Calculating and comparing volumes of rectangular prisms and right circular cylinders in terms of algebraic variables.

Soru

Cevap

Adım Adım Çözüm

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