Exercise 34 Page 176

Practice makes perfect

a We have been given a table showing the cost for renting a beach house.

Days	$2$	$4$	$6$	$8$
Total Cost (dollars)	$246$	$450$	$546$	$858$

The situation can be modeled by a linear equation if the rate of change is constant. We can calculate the rate of change using the Slope Formula.

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

Let's find the rate of change for the ordered pairs in the table.

$(x_{1}, y_{1}),$ $(x_{2}, y_{2})$	$\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	$m$
$(2, 246),$ $(4, 450)$	$\frac{4 5 0 - 2 4 6}{4 - 2}$	$102$
$(4, 450),$ $(6, 546)$	$\frac{5 4 6 - 4 5 0}{6 - 4}$	$102$
$(6, 546),$ $(8, 858)$	$\frac{8 5 8 - 5 4 6}{8 - 6}$	$102$

We can see that the rate of change is constant. Therefore, the situation can be modeled by a linear equation.

b To determine the processing fee and the daily fee, let's find the linear equation which models this situation. Recall the point-slope form of a linear equation.

y - y_{1} = m (x - x_{1})

Here

m

is the rate of change and

(x_{1}, y_{1})

is a point on the line. In Part A we already determined that the rate of change is

102 .

Let's substitute this and the point

(2, 246)

into the formula.

y - y_{1} y - 246 = m (x - x_{1}) ⇓ = 102 (x - 2)

To find the processing fee and the daily fee we need to rewrite this into slope-intercept form.

y - 246 = 102 (x - 2)

▼

Solve for

y

Distr

Distribute $102$

y - 246 = 102 x - 204

AddEqn

$LHS + 246 = RHS + 246$

y = 102 x + 42

We now have an equation which models the situation.

y = 102 x + 42

Now we know the daily fee is

$ 102 .

The processing fee is

$ 42,

as this is a fixed amount that is paid only once no matter how many days the beach house is rented.

c Since we can spend no more than

$ 1200

on the beach house rental, the cost must be less than or equal to

1200 .

We can represent this as an inequality.

102 x + 42 \leq 1200

Let's solve it!

102 x + 42 \leq 1200

▼

Solve for

x

SubIneq

$LHS - 42 \leq RHS - 42$

102 x \leq 1158

DivIneq

$LHS / 102 \leq RHS / 102$

x \leq \frac{1 1 5 8}{1 0 2}

UseCalc

Use a calculator

x \leq 11.35294 \dots

RoundDec

Round to $1$ decimal place(s)

x \leq 11.4

Since

x

is the number of days and we cannot rent the house for a fraction of a day, we can conclude that the maximum number of days we can rent the beach house is

11 .

Subchapter links

Communicate Your Answer p.171

Monitoring Progress p.172-174

Exercises p.175-176

Exercise 34 Page 176

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