Question

Marguerite wants to rent a carpet cleaner. Company A rents a carpet cleaner for $15 per day. Company B rents a carpet cleaner for $100 per week, with a one-time fee of $5.

The following functions represent the rate structures of the two rental companies:

x = the number of weeks

Company A

f(x) = 15(7x)

Company B

g(x) = 100x + 5

The function h(x) = f(x) – g(x) represents the difference between the two rate structures.

Determine which statements about h(x) and about renting a carpet cleaner are true. Check all that apply.
If Marguerite rents for 2 weeks, it will cost her more if she rents from Company B.
If Marguerite rents for 2 weeks, it will cost her less if she rents from Company B.
If Marguerite rents for 1 week, it will cost her the same at either company.
If Marguerite rents for 1 week, it will cost her more if she rents from Company A.
h(x) = 5x – 5
h(x) = 5x + 5

Answer 1

Explanation:

h(x) = f(x) - g(x) = 15(7x) - (100x + 5) =

= 105x - 100x - 5 = 5x - 5

if h(x) < 0, then g(x) (company B) is more expensive.

if h(x) > 0, then f(x) (company A) is more expensive.

if h(x) = 0, then both companies charge the same.

2 weeks :

h(2) = 5×2 - 5 = 10 - 5 = 5

company A is more expensive, so, it costs her less to rent from company B.

1 week :

h(1) = 5×1 - 5 = 5 - 5 = 0

it will cost her the same with either company.

Answer 2

If Marguerite rents for 2 weeks, it will cost her less if she rents from Company B.

If Marguerite rents for 1 week, it will cost her the same at either company

h(x) = 5x - 5

Explanation:

Given information:

• Company A rents a carpet cleaner for $15 per day.
• Company B rents a carpet cleaner for $100 per week, with a one-time fee of $5.

Let x = number of weeks

Company A rate structure:

`f(x)=15(7x)`

Company B rate structure:

`g(x)=100x+5`

2 week rental

`\textsf{Company A}:\quad f(2)=15(7\cdot 2)=210`

`\textsf{Company B}:\quad g(2)=100(2)+5=205`

Therefore, it will cost less renting from Company B.

1 week rental

`\textsf{Company A}:\quad f(1)=15(7\cdot 1)=105`

`\textsf{Company B}:\quad g(1)=100(1)+5=105`

Therefore, it will cost the same at either company.

Difference between two rate structures

`\begin{aligned}\textsf{If }h(x) & = f(x)-g(x)\\\\\implies h(x) & = 15(7x)-(100x+5)\\& = 105x-100x-5\\& = 5x-5\end{aligned}`

True Statements

• If Marguerite rents for 2 weeks, it will cost her less if she rents from Company B.
• If Marguerite rents for 1 week, it will cost her the same at either company
• h(x) = 5x - 5