Q10. - Algebra 1

Question

The table shows the total cost of renting a canoe for $n$ hours.

Number of Hours $\left(n\right)$	Rental Cost $\left(c\right)$
1	\(15
2	\)20
3	\(25
4	\)30

a. Write a function to represent the situation.

b. How much would it cost to rent the canoe for 7 hours?

Step-by-Step Solution

Answer

a. The function to represent the situation is $\mathit{C}\mathbf{=}\mathbf{5}\mathit{n}\mathbf{+}\mathbf{10}$

b. The cost to rent the canoe for 7 hours is $45.

Part a. Step 1. Calculate the difference of successive c -values.

The difference of successive $C$-values is;

 From the above difference of successive $C$-values, it can be noticed that the first difference of successive $C$-values is equal. Therefore, the function should be a linear function.

Part a. Step 2. Write a function to represent the situation .

As, the first difference of successive $C$-values is equal. Therefore, the function should be a linear function.

Therefore, the equation of the function will be of the form $C=mn+a$, where $C$ is the rental cost and $n$ is the number of hours.

As, it is given that when $n=1,C=15$.

Therefore, 

$$\begin{gathered} C=mn+a \\ 15=m\left(1\right)+a \\ 15=m+a\left(1\right) \end{gathered}$$

As, it is given that when $n=2,C=20$.

Therefore,

$$\begin{gathered} C=mn+a \\ 20=m\left(2\right)+a \\ 20=2m+a\left(2\right) \end{gathered}$$

Subtract the equation (1) from the equation (2).

$$\begin{gathered} 20-15=2m+a-\left(m+a\right) \\ 5=2m+a-m-a \\ 5=m \end{gathered}$$

Substitute the value of $m$ in the equation (1).

      $$\begin{gathered} 15=m+a \\ 15=5+a \end{gathered}$$

$$\begin{gathered} 15-5=a \\ 10=a \end{gathered}$$

Therefore, the values of $m$ and $a$ are 5 and 10 respectively.

Substitute the values of $m$ and $a$ in the equation $C=mn+a$.

$$\begin{gathered} C=\left(5\right)n+10 \\ C=5n+10 \end{gathered}$$

Therefore, the function to represent the situation is $C=5n+10$.

Part b. Step 1. Write the equation of the function that relates to the rental cost and the number of hours.

The equation of the function that relates the rental cost and the number of hours is $C=5n+10$.

Part b. Step 2. Find how much would it cost to rent the canoe for 7 hours.

It is given that the value of $n$ is 7.

Substitute 7 for $n$ in the equation $C=5n+10$ to find the value of $C$.

$$\begin{gathered} C=5\left(7\right)+10 \\ =35+10 \\ =45 \end{gathered}$$

Therefore, the cost to rent the canoe for 7 hours is $45.