Chapter 4.2, Problem 40E

a.

To determine

The sibling who received the most money and the one who received least money.

Answer to Problem 40E

Sibling B received the most money ($125) and siblings A received the least money ($80).

Explanation of Solution

Given Information: Graph, table and equation for respective siblings where $x$ is week and $y$ is money left.

Calculation:

For sibling $A$ :

Point slope form: $y-y_1=m(x-x_1)$

  $m=\frac{y-y_1}{x-x_1}$

Considering the points $(2,50)$ and $(4,20)$ ,

  $\begin{array}{l}m=\frac{20-50}{4-2}\\ m=\frac{-30}{2}\\ m=-15\end{array}$

Substituting the values in point slope form,

  $\begin{array}{l}y-20=-15(x-4)\\ y=-15x+60+20\\ y=-15x+80\end{array}$ (Distributive property and rearranging)

When $x=0$ ,

  $y=80$

Hence, sibling $A$ received $\$80$ .

For sibling $B$ :

Point slope form: $y-y_1=m(x-x_1)$

  $m=\frac{y-y_1}{x-x_1}$

Considering the points $(1,100)$ and $(4,25)$ ,

  $\begin{array}{l}m=\frac{25-100}{4-1}\\ m=\frac{-75}{3}\\ m=-25\end{array}$

Substituting the values in point slope formula,

  $\begin{array}{l}y-25=-25(x-4)\\ y=-25x+100+25\\ y=-25x+125\end{array}$ (Distributive property and rearranging)

When $x=0$ ,

  $y=125$

Hence, sibling $B$ received $\text{\$125}$ .

For sibling $\text{C}$ :

  $y=-22.5x+90$

When $x=0$ ,

  $y=90$

Hence, sibling $\text{C}$ received $\$90$ .

Sibling B received the most money ($125) and siblings A received the least money ($80).

b.

To determine

To find: Sibling that spends money at fastest rate.

Answer to Problem 40E

Siblings B spends at the fastest rate that is $25 per week and siblings A spends the money at lowest rate that is $15 per week.

Explanation of Solution

Given information: Graph, table and equation for respective siblings where $x$ is week and $y$ is money left.

Calculation:

As calculated in previous part,

  $\begin{array}{l}\text{For sibling A,}\\ m=-15\\ \text{For sibling B,}\\ m=-25\\ \text{For sibling C,}\\ m=-22.5\end{array}$

Thus, siblings B spends at the fastest rate that is $25 per week and siblings A spends the money at lowest rate that is $15 per week.

c.

To determine

To find: The sibling that runs out of money first and the one that runs out the money at the last.

Answer to Problem 40E

Siblings C exhausts money first (in 4 weeks) and siblings A exhausts money last (in 5.33 weeks).

Explanation of Solution

Given information: Graph, table and equation for respective siblings where $x$ is week and $y$ is money left.

Calculation:

Putting $y=0$ in sibling $A\text{'}s$ equation,

  $0=-15x+80$

  $\begin{array}{l}x=\frac{16}{3}\\ x\approx 5.33\mathrm{weeks}\end{array}$

Putting $y=0$ in sibling $B\text{'}s$ equation,

  $\begin{array}{l}0=-25x+125\\ x=5\mathrm{weeks}\end{array}$

Putting $y=0$ in sibling $C\text{'}s$ equation,

  $\begin{array}{l}0=-22.5x+90\\ x=4\mathrm{weeks}\end{array}$

Siblings C exhausts money first (in 4 weeks) and siblings A exhausts money last (in 5.33 weeks).