Exercise 52 Page 535

Practice makes perfect

a The base of a rectangular prism is always a rectangle. Therefore, to find the length of

\overline{B C}

we will use the formula for the area of a rectangle.

Area = ℓ \cdot w

In our case we can assume that

ℓ = A B

and

w = B C .

Area = A B \cdot B C

Next, let's consider the given information. We know that the area of the base of the prism is equal to

20 cm^{2} .

From the diagram we have that

A B = x .

Using this information and the formula above, we can write

B C

in terms of

x .

Let's do it!

Area = A B \cdot B C

SubstituteII

$Area = 20$ , $A B = x$

20 = x \cdot B C

DivEqn

$LHS / x = RHS / x$

\frac{2 0}{x} = B C

RearrangeEqn

Rearrange equation

B C = \frac{2 0}{x}

The length of

\overline{B C}

is equal to

\frac{2 0}{x}

centimeters.

b We want to find the area of the shaded region.

Since the shaded region is a side wall of a rectangular prism, it is a rectangle. Therefore, to calculate its area we will use the formula for the area of a rectangle.

Area = ℓ \cdot w

In our case we can assume that

ℓ = D C

and

w = B C .

Area = D C \cdot B C

We are also given that

D C = 3 B C .

Let's substitute this expression into the formula above and simplify.

Area = 3 B C \cdot B C \Leftrightarrow Area = 3 B C^{2}

Finally, we want to express the area of the shaded region in terms of

x .

From Part A we know that

B C = \frac{2 0}{x} cm .

Area = 3 B C^{2}

Substitute

$B C = \frac{2 0}{x}$

Area = 3 (\frac{2 0}{x})^{2}

▼

Simplify

PowQuot

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

Area = 3 \cdot \frac{2 0 ^{2}}{x ^{2}}

CalcPow

Calculate power

Area = 3 \cdot \frac{4 0 0}{x ^{2}}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

Area = \frac{1 2 0 0}{x ^{2}}

The area of the shaded region is equal to

\frac{1 2 0 0}{x ^{2}}

square centimeters.

c We will use the formula for the volume of a prism.

Volume = Area of the Base \cdot Height

We are told that the area of the base is

20 cm^{2} .

Volume = 20 \cdot Height

The height of the given rectangular prism is equal to

D C .

From Part B we know that

D C = 3 B C

and from Part A we know that

B C = \frac{2 0}{x} .

Let's bring all of this information together.

Height = D C = 3 B C = 3 (\frac{2 0}{x}) = \frac{6 0}{x}

Finally, we can calculate the volume of the prism.

Volume = 20 \cdot \frac{6 0}{x} = \frac{1 2 0 0}{x}

The volume of the prism is

\frac{1 2 0 0}{x}

cubic centimeters.

Subchapter links

Check Your Understanding p.533

Practice and Problem Solving p.534-536

H.O.T. Problems p.536

Standardized Test Practice p.537

Spiral Review p.537

Skills Review p.537

Exercise 52 Page 535

Hints

Check the answer

Practice exercises

Asses your skill