Use the information given to answer the question.

Question

A family rents a personal watercraft and a speedboat for use on a lake.

To rent the personal watercraft, the family pays an $80
security deposit in addition to a rental fee of $42
per hour.
To rent the speedboat, the family pays a $200
security deposit in addition to a rental fee of $60
per hour.
Part A

If the total cost for the family to rent the personal watercraft and speedboat for the same number of hours is $790
, which equation could be used to determine the number of hours,
, the family rents both the personal watercraft and the speedboat?

Responses

102 h + 280 = 790

102 h + 220 = 790

18 h + 280 = 790

Answer 1

To determine the equation that represents the total cost of renting both the personal watercraft and the speedboat, let's break down the costs for each rental based on the provided information.

1. Personal Watercraft:

   • Security deposit: $80
   • Rental fee: $42 per hour
   • Total cost for \( h \) hours: \( 80 + 42h \)

2. Speedboat:

   • Security deposit: $200
   • Rental fee: $60 per hour
   • Total cost for \( h \) hours: \( 200 + 60h \)

Now, we add the total costs for both rentals together:

\[ (80 + 42h) + (200 + 60h) \]

This simplifies to:

\[ 80 + 200 + 42h + 60h = 280 + 102h \]

According to the problem, the total cost for renting both the personal watercraft and the speedboat is $790. Therefore, we can set up the equation:

\[ 280 + 102h = 790 \]

Now, let's compare this to the options provided:

1. \( 102h + 280 = 790 \) (This is equivalent)
2. \( 102h + 220 = 790 \) (Incorrect, does not match our derived equation)
3. \( 18h + 280 = 790 \) (Incorrect, does not match our derived equation)

The correct equation that can be used to determine the number of hours \( h \) is:

102 h + 280 = 790

Answer 2

Part B

A second family is deciding whether to rent a personal watercraft or a speedboat for use on the lake for
hours. Which statement is true?

Responses

The family will save exactly
by renting the personal watercraft instead of the speedboat.

The family will save exactly $ 138 by renting the personal watercraft instead of the speedboat.

The family will save exactly 138
by renting the speedboat instead of the personal watercraft.

The family will save exactly $ 138 by renting the speedboat instead of the personal watercraft.

The family will save exactly 210
by renting the personal watercraft instead of the speedboat.

The family will save exactly $ 210 by renting the personal watercraft instead of the speedboat.

Answer 3

To determine how much a family will save by renting either a personal watercraft or a speedboat for \( h \) hours, we need to calculate the total costs for both options and find the difference.

1. Cost of renting a personal watercraft for \( h \) hours: \[ \text{Cost}_{\text{PWC}} = 80 + 42h \]

2. Cost of renting a speedboat for \( h \) hours: \[ \text{Cost}_{\text{Speedboat}} = 200 + 60h \]

Now, let's find the difference in cost between renting the personal watercraft and the speedboat:

\[ \text{Savings} = \text{Cost}{\text{Speedboat}} - \text{Cost}{\text{PWC}} = (200 + 60h) - (80 + 42h) \]

This simplifies to:

\[ \text{Savings} = 200 + 60h - 80 - 42h = 120 + 18h \]

Now we consider the different statements provided:

1. The family will save exactly [amount] by renting the personal watercraft instead of the speedboat.

   • This will only be true if the savings calculation results in a positive value.

2. The family will save exactly $ 138 by renting the personal watercraft instead of the speedboat.

   • Set \( 120 + 18h = 138 \) to find \( h \): \[ 18h = 138 - 120 \quad \to \quad 18h = 18 \quad \to \quad h = 1 \]

3. The family will save exactly 138 by renting the speedboat instead of the personal watercraft.

   • This would imply the cost of the speedboat is less, which is not possible since we've shown the opposite.

4. The family will save exactly $ 138 by renting the speedboat instead of the personal watercraft.

   • This is impossible since the speedboat has a higher cost based on the previously derived savings equation.

5. The family will save exactly 210 by renting the personal watercraft instead of the speedboat.

   • Set \( 120 + 18h = 210 \) to find \( h \): \[ 18h = 210 - 120 \quad \to \quad 18h = 90 \quad \to \quad h = 5 \]

6. The family will save exactly $ 210 by renting the personal watercraft instead of the speedboat.

   • This implies the savings are equal to that amount, which we found possible based on \( h = 5 \).

Thus, there are two possible savings but considering all options, the most relevant to the derivation is:

The family will save exactly $ 138 by renting the personal watercraft instead of the speedboat (when \( h = 1 \)).

So the correct answer is the second option: The family will save exactly $ 138 by renting the personal watercraft instead of the speedboat.