Exercise 40 Page 176

Practice makes perfect

a The amount of money each sibling is given is equal to the

y -

intercept of the function that models their spending behavior. This is because when they receive their money nothing has been spent yet. The time is then

x = 0 .

Sibling A

Let's consider the graph that illustrates how Sibling A spends the money.

Graph of a linear function that passes through the points (2,50) and (4,20) showing the y-intercept at y=80

From the graph, we see that the $y -$ intercept occurs at $(0, 80),$ which means this sibling received $$ 80 .$

Sibling B

Let's view the function that tells how Sibling B spends money.

y = - 22.5 x + 90

From the function, we see that the

y -

intercept occurs at

(0, 90) .

This sibling received

$ 90 .

Sibling C

In the table we do not have an ordered pair where

x = 0 .

But since the sibling spends the money at a constant rate, we can use the rate of change to find the

y -

intercept. We will use the Slope Formula and the points

(1, 100)

and

(2, 75)

to calculate the rate of change.

Rate of change = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(1, 100)$ & $(2, 75)$

Rate of change = \frac{7 5 - 1 0 0}{2 - 1}

▼

Simplify right-hand side

SubTerms

Subtract terms

Rate of change = \frac{- 2 5}{1}

DivByOne

$\frac{a}{1} = a$

Rate of change = - 25

Sibling C has a rate of change of

- 25,

which corresponds to spending

$ 25

each week. After one week of spending the sibling had

$ 100

left. This means that the sibling must have started with

$ 125 .

Summarize

Now we can summarize and determine who received what amount of money.

Sibling A : Sibling B : Sibling C : $ 80 (l e a s t) $ 90 $ 125 (m o s t)

b Comparing the spending rates means comparing the slopes of the spending functions. Let's do it!

Sibling A

To find the slope for Sibling A we can use the points from the graph,

(2, 50)

and

(4, 20),

and substitute them into the Slope Formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(2, 50)$ & $(4, 20)$

m = \frac{2 0 - 5 0}{4 - 2}

SubTerms

Subtract terms

m = \frac{- 3 0}{2}

CalcQuot

Calculate quotient

m = - 15

A slope of

- 15

means that Sibling A spends their money at a rate of

$ 15

per week.

Sibling B

For Sibling B, we have a function where we can identify the slope by looking at the coefficient of

x .

y = - 22.5 x + 90

The slope for Sibling B is

- 22.5,

indicating a spending rate of

$ 22.50

per week.

Sibling C

In Part A we determined the rate of change for Sibling C to be $- 25 .$ Thus, the spending rate is $$ 25$ per week.

Summarize

Let's summarize what we have found.

Sibling A : Sibling B : Sibling C : $ 15 a week (l e a s t) $ 22.50 a week $ 25 a week (m o s t)

c We can determine when all of the money has been spent by finding the

x -

intercept.

Sibling A

Sibling A spends

$ 15

a week and starts out with

$ 80 .

This means we can model their spending behavior with a linear function with a slope of

- 15

and a

y -

intercept of

80 .

y = - 15 x + 80

We find the

x -

intercept by substituting

0

for

y

and solving for

x .

y = - 15 x + 80

Substitute

$y = 0$

0 = - 15 x + 80

▼

Solve for

x

AddEqn

$LHS + 15 x = RHS + 15 x$

15 x = 80

DivEqn

$LHS / 15 = RHS / 15$

x = 5.33333 \dots

RoundDec

Round to $1$ decimal place(s)

x \approx 5.3

Sibling A runs out of money after about

5.3

weeks.

Sibling B

We know that Sibling B has $$ 25$ left at the end of $4$ weeks. Since they spend $$ 25$ a week, the remaining money will last one more week. Thus, Sibling B runs out of money after $5$ weeks.

Sibling C

We have a linear function that tells us how much money Sibling C has at a given time.

y = - 22.5 x + 90

We find the

x -

intercept by substituting

0

for

y

and solving for

x .

y = - 22.5 x + 90

Substitute

$y = 0$

0 = - 22.5 x + 90

▼

Solve for

x

AddEqn

$LHS + 22.5 x = RHS + 22.5 x$

22.5 x = 90

DivEqn

$LHS / 22.5 = RHS / 22.5$

x = 4

Sibling C runs out of money at the end of week

4 .

Summarize

Let's summarize when each of the siblings run out of money.

Sibling A : Sibling B : Sibling C : 5.3 weeks (l a s t) 5 weeks 4 weeks (f i r s t)

Subchapter links

Communicate Your Answer p.171

Monitoring Progress p.172-174

Exercises p.175-176

Exercise 40 Page 176

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Check the answer

Practice exercises

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Sibling A

Sibling B

Sibling C

Summarize

Sibling A

Sibling B

Sibling C

Summarize

Sibling A

Sibling B

Sibling C

Summarize