Chapter 1.5, Problem 3MRPS

Solution

a.

To write: An expression for $w$ in terms of $h$ .

An expression for $w$ in terms of $h$ is $w=\frac{4}{3}h$

Given:

The diagonal of the screen on a laptop computer measures 15 inches. The ratio of the screen's width $w$ to its height $h$ is $4:3$ .

Calculations:

The ratio of $w$ to $h$ is given as $4:3$ . Thus, $\frac{w}{h}=\frac{4}{3}$

Find the cross products and solve for $w$ .

  $\begin{array}{c}w\cdot 3=4\cdot h\\ \frac{w\cdot 3}{3}=\frac{4\cdot h}{3}\\ w=\frac{4}{3}h\end{array}$

b.

To write: An equation that you can use to find $h$ .

An equation that you can use to find $h$ is $h^2=81$ .

Given:

The diagonal of the screen on a laptop computer measures 15 inches. The ratio of the screen's width $w$ to its height $h$ is $4:3$ .

Calculations:

Consider the following figure.

  

Apply the Pythagorean Theorem.

  $h^2+w^2={15}^2$

Substitute for $w$ obtained in part a.

  $\begin{array}{c}{h}^2+{\left(\frac{4}{3}h\right)}^2=225\\ h^2+\frac{16}{9}{h}^2=225\end{array}$

Simplify to get an equation for $h$ .

  $\begin{array}{c}\frac{25}{9}{h}^2=225\\ \frac{25}{9}{h}^2\cdot \frac{9}{25}=225\cdot \frac{9}{25}\\ h^2=81\end{array}$

c.

To solve: An equation from part (b).

Solution is $9$ .

Given:

The diagonal of the screen on a laptop computer measures 15 inches. The ratio of the screen's width $w$ to its height $h$ is $4:3$ .

Calculations:

For solving the equation, take the square root on each side.

  $\begin{array}{c}\sqrt{h^2}=\pm \sqrt{81}\\ h=\pm 9\end{array}$

Since height cannot be negative, reject the solution $h=-9$ . Thus, the possible solution is $9$ .

d.

To find: Height, width, and area of the laptop screen.

The height of the laptop screen is $9$ inches, the width of the screen is $12$ inches and the area of the screen is $108$ square inches.

Given:

The diagonal of the screen on a laptop computer measures 15 inches. The ratio of the screen's width $w$ to its height $h$ is $4:3$ .

Calculations:

The height of the laptop screen is $9$ inches.

For finding the width, substitute $9$ for $h$ in the expression for $w$ and evaluate.

  $\begin{array}{c}w=\frac{4}{3}\left(9\right)\\ =12\end{array}$

The width of the screen is $12$ inches.

Since the screen is in the shape of a rectangle, its area is the product of the width and the height.

  $\begin{array}{c}\text{Area}=9\left(12\right)\\ =108\end{array}$

Therefore, the area of the screen is $108$ square inches.