Chapter 3.4, Problem 42PPS

(a)

To determine

Write an equation to represent V’s savings y after x weeks.

Answer to Problem 42PPS

y = 15x + 50

Explanation of Solution

Given:

V is saving his money to buy a new satellite radio for his car.He wants to save enough money for the radio and one year of satellite radio service before he buys it .He started saving for the radio with $50 that he got for his birthday.Since then , he has been adding $15 each week after he cashes his paycheck.

Calculation :

The savings of V = y

Number of weeks = x

Since he saves $15 each week , so , he saves in x weeks = 15x

Initially , he saved $50

So, the total savings after x weeks = 15x + 50

The equation that represents V’s savings y after x weeks :

y = 15x + 50

(b)

To determine

Graph the equation is part (a)

Explanation of Solution

Calculation :

From part (a)

The equation that represents V’s savings y after x weeks :

y = 15x + 50

Make a table to locate the coordinates of the equation:

x	y = 15x + 50
0	50
10	200

Now, plot the points and join them to draw the graph of the equation y = 15x + 50:

  

(c)

To determine

Find the number of weeks taken by V to save $150.

Answer to Problem 42PPS

After 7 weeks , V will have atleast $150.

Explanation of Solution

Calculation :

From part (a),

The equation that represents V’s savings y after x weeks :

y = 15x + 50

where

The savings of V = y = $150 (given)

Number of weeks = x

Substitute the value of y in the equation and solve for x :

  $\begin{array}{l}y=15x+50\\ \Rightarrow 150=15x+50\\ \Rightarrow 15x=150-\mathrm{50...........}[\text{subtract 50 from each side]}\\ \Rightarrow x=\frac{100}{15}\mathrm{.....................}[\text{divide each side by 15]}\\ \Rightarrow x=6.6666\end{array}$

So, after 7 weeks , V will have atleast $150.

(d)

To determine

Find the number of weeks taken by V to save enough money.

Answer to Problem 42PPS

It will take about 12 weeks to save enough money.

Explanation of Solution

Given:

A satellite radio costs $180 .Satellite radio service costs $10 per month.V started saving two weeks ago.

Calculation :

From part (a),

The equation that represents V’s savings y after x weeks :

y = 15x + 50

where

The savings of V = y = $150 (given)

Number of weeks = x

He wants to pay for the radio and 1 year of service, so , total amount required by V is :

  $180+12\times 10=\$300$ , since the cost of service of 1 month is $10 and there are 12 months in 1 year.

He is saving for two weeks now , so, the amount V has already saved :

  $\begin{array}{l}y=15x+50\\ \Rightarrow y=15(2)+50\\ \Rightarrow y=30+50\\ \Rightarrow y=\$80\end{array}$

So, he has $80 with him already.

Now, he needs to save = $300 - $80 = $220

So, substitute y = 220 in the equation to find the number of weeks:

  $\begin{array}{l}y=15x+50\\ \Rightarrow 220=15x+50\\ \Rightarrow 15x=220-\mathrm{50................}[\text{subtract 50 from each side]}\\ \Rightarrow 15x=170\\ \Rightarrow x=\frac{170}{15}\mathrm{..........................}[\text{divide each side by 15]}\\ \Rightarrow x=11.3333\end{array}$

So, it will take about 12 weeks to save enough money.