Chapter 1.3, Problem 124AYU

U.S. Advertising Share A report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads (magazines and newspapers) accounted for $26\%$ of the spending. The report further showed that Internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Write a linear equation that relates that percent $y$ of print ad spending to the percent $x$ of Internet ad spending.

Find the intercepts of the graph of your equation.

Do the intercepts have any meaningful interpretation?

Predict the percent of print ad spending if Internet ads account for $39\%$ of all advertisement spending in the United States.

To determine

A linear equation that relates that percent $y$ of print ad spending to the percent $x$ of internet ad spending, where a report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads accounted for $26\%$ of the spending. The report further showed that internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Answer to Problem 124AYU

Solution:

A linear equation that relates that percent $y$ of print ad spending to the percent $x$ of internet ad spending is $y=-\frac{5}{8}x+\frac{303}{8}$.

Explanation of Solution

Given Information:

A report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads accounted for $26\%$ of the spending. The report further showed that internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Explanation:

Let, $y$ be percent of print ad spending and $x$ be percent of internet ad spending.

If print ads accounted for $26\%$, Internet ads accounted for $19\%$.

When $x=19$, $y=26$.

In ordered pair it can be written as $\left(19,26\right)$.

If print ads accounted for $16\%$, Internet ads accounted for $35\%$.

When $x=35$, $y=16$.

In ordered pair it can be written as $\left(35,16\right)$.

Slope of the linear equation passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$is $m=\frac{y_2-y_1}{x_2-x_1}$.

$\Rightarrow m=\frac{16-26}{35-19}=\frac{-10}{16}=\frac{-5}{8}$

The general form linear equation is $y=mx+b$ where $m$ is a slope and $b$ is $y-$ intercept.

Substitute $m=\frac{-5}{8}$,

$\Rightarrow y=-\frac{5}{8}x+b$

To find $y-$ intercept substitute $x=35$ and $y=16$.

$\Rightarrow 16=-\frac{5}{8}\cdot 35+b$

$\Rightarrow 16=-\frac{175}{8}+b$

Adding $\frac{175}{8}$ from both sides,

$\Rightarrow b=\frac{175}{8}+16$

$\Rightarrow b=\frac{303}{8}$

$\Rightarrow y=-\frac{5}{8}x+\frac{303}{8}$

Therefore, a linear equation that relates that percent $y$ of print ad spending to the percent $x$ of internet ad spending is $y=-\frac{5}{8}x+\frac{303}{8}$.

To determine

The intercept of the graph of equation that relates that percent $y$ of print ad spending to the percent $x$ of internet ad spending.

Answer to Problem 124AYU

Solution:

The $x-$ intercept of the graph of equation $y=-\frac{5}{8}x+\frac{303}{8}$ is $x=\frac{303}{5}$ and the $y-$ intercept is $y=\frac{303}{8}$.

Explanation of Solution

Given Information:

A report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads accounted for $26\%$ of the spending. The report further showed that internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Explanation:

From the part (a), the linear equation is $y=-\frac{5}{8}x+\frac{303}{8}$.

To find $x-$ intercept set $y=0$.

$\Rightarrow -\frac{5}{8}x+\frac{303}{8}=0$

$\Rightarrow \frac{5}{8}x=\frac{303}{8}$

Multiply both sides by 8,

$\Rightarrow 5x=303$

$\Rightarrow x=\frac{303}{5}$

The $x-$ intercept of the graph of equation $y=-\frac{5}{8}x+\frac{303}{8}$ is $x=\frac{303}{5}$.

To find $y-$ intercept set $x=0$.

$\Rightarrow y=-\frac{5}{8}\cdot 0+\frac{303}{8}$

$\Rightarrow y=\frac{303}{8}$

The $y-$ intercept of the graph of equation $y=-\frac{5}{8}x+\frac{303}{8}$ is $y=\frac{303}{8}$.

To determine

Whether theintercept have any meaningful interpretation or not.

Answer to Problem 124AYU

Solution:

Yes.

Explanation of Solution

Given Information:

A report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads accounted for $26\%$ of the spending. The report further showed that internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Explanation:

To find $x-$ intercept set $y=0$.

From the part (b), the $x-$ intercept of the graph of equation $y=-\frac{5}{8}x+\frac{303}{8}$ is $x=\frac{303}{5}$.

To find $y-$ intercept set $x=0$.

From the part (b), the $y-$ intercept of the graph of equation $y=-\frac{5}{8}x+\frac{303}{8}$ is $y=\frac{303}{8}$.

Let, $y$ be percent of print ad spending and $x$ be percent of internet ad spending.

Here, the both intercept are positive.

Since the percent of advertisement should not be negative.

Therefore, the intercept have meaningful interpretation.

To determine

The percent of print ad spending if Internet ads account for $39\%$ of all advertisement spending in the United States.

Answer to Problem 124AYU

Solution:

The percent of print ad spending is $13.5\%$ if internet ads account for $39\%$ of all advertisement spending in the United States.

Explanation of Solution

Given Information:

A report showed that Internet ads accounted for $19\%$ of all U.S. advertisement spending when print ads accounted for $26\%$ of the spending. The report further showed that internet ads accounted for $35\%$ of all advertisement spending when print ads accounted for $16\%$ of the spending.

Explanation:

Let, $y$ be percent of print ad spending and $x$ be percent of internet ad spending.

From the part (a), linear equation is $y=-\frac{5}{8}x+\frac{303}{8}$.

To find the percent of print ad spending if Internet ads account for $39\%$, substitute $x=39$ in $y=-\frac{5}{8}x+\frac{303}{8}$ and find $y$.

$\Rightarrow y=-\frac{5}{8}\cdot 39+\frac{303}{8}$

$\Rightarrow y=-\frac{195}{8}+\frac{303}{8}$

$\Rightarrow y=\frac{303-195}{8}$

$\Rightarrow y=\frac{108}{8}$

$\Rightarrow y=13.5\%$

Therefore, the percent of print ad spending is $13.5\%$ if Internet ads account for $39\%$ of all advertisement spending in the United States.