Q51. - Algebra 1

Question

An amusement park offers a yearly membership of \(275 that allows for free parking and admission to the park. Members can also use the water park for an additional \)5 per day. Nonmember pay \(6 for parking, \)15 for admission, and $9 for the water park.

 

1. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit.
2. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park.
3. Plot these point on a coordinate graph and describe things you notice from the graph.

Step-by-Step Solution

Answer

1. The number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit is 11.
2. The required table is shown below:

Visits	Cost for Members ($)	Cost for nonmembers ($)
3	290	90
6	305	180
9	320	270
12	325	360
15	350	450

c. The graph is shown below:

From the graph, it can be observed that the costs of a member increased by $5 per visit and the costs of nonmembers increased by $30 per visit. 

a. Step 1. Make the equation.

Let the number of visits is represented by x.

Member can use the water park for an additional $5 per day.

Thus, the amount spent by the member for x visits is: $275+5x$

The amount spent by the Nonmembers for x visits is: $6x+15x+9x$

It is given that the total cost for members and Nonmembers should be the same.

Thus, the required equation is: $6x+15x+9x=275+5x$

Step 2. Isolate the variable.

$\begin{array}{l}30x=275+5x\\ 30x-5x=275\\ 25x=275\end{array}$

Step 3. Divide both the sides by 25.

$\begin{array}{l}\frac{25x}{25}=\frac{275}{25}\\ x=11\end{array}$

Hence, the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit is 11.

b. Step 1. Cost of members and nonmembers after 3 visits.

We know that amount spent by the member for x visits is $275+5x$ and for Nonmembers is $6x+15x+9x$.

Substitute 3 for x into $275+5x$.

$\begin{array}{c}275+5\left(3\right)=275+15\\ =290\end{array}$

Substitute 3 for x into $6x+15x+9x$.

$\begin{array}{c}6x+15x+9x=6\left(3\right)+15\left(3\right)+9\left(3\right)\\ =18+45+27\\ =90\end{array}$

Step 2. Cost of members and nonmembers after 6 visits.

We know that amount spent by the member for x visits is $275+5x$ and for Nonmembers is $6x+15x+9x$.

Substitute 6 for x into $275+5x$.

$\begin{array}{c}275+5\left(6\right)=275+30\\ =305\end{array}$

Substitute 6 for x into $6x+15x+9x$.

$\begin{array}{c}6x+15x+9x=6\left(6\right)+15\left(6\right)+9\left(6\right)\\ =180\end{array}$

Step 3. Cost of members and nonmembers after 9 visits.

We know that amount spent by the member for x visits is $275+5x$ and for Nonmembers is $6x+15x+9x$.

Substitute 9 for x into $275+5x$.

$\begin{array}{c}275+5\left(9\right)=275+45\\ =320\end{array}$

Substitute 9 for x into $6x+15x+9x$.

$\begin{array}{c}6x+15x+9x=6\left(9\right)+15\left(9\right)+9\left(9\right)\\ =270\end{array}$

Step 4. Cost of members and nonmembers after 12 visits.

We know that amount spent by the member for x visits is $275+5x$ and for Nonmembers is $6x+15x+9x$.

Substitute 12 for x into $275+5x$.

$\begin{array}{c}275+5\left(12\right)=275+60\\ =335\end{array}$

Substitute 12 for x into $275+5x$.

$\begin{array}{c}6x+15x+9x=6\left(12\right)+15\left(12\right)+9\left(12\right)\\ =360\end{array}$

Step 5. Cost of members and nonmembers after 15 visits.

We know that amount spent by the member for x visits is $275+5x$ and for Nonmembers is $6x+15x+9x$.

Substitute 15 for x into $275+5x$.

$\begin{array}{c}275+5\left(15\right)=275+75\\ =350\end{array}$

Substitute 15 for x into $6x+15x+9x$.

$\begin{array}{c}6x+15x+9x=6\left(15\right)+15\left(15\right)+9\left(15\right)\\ =450\end{array}$

Hence, the required table is:

Visits	Cost for Members ($)	Cost for nonmembers ($)
3	290	90
6	305	180
9	320	270
12	335	360
15	350	450

c. Step 1. Make the ordered pairs.

From the above table the ordered pairs are:

For member: $(3,290),(6,305),(9,320),(12,335),(15,350)$

For nonmember: $(3,90),(6,180),(9,270),(12,360),(15,450)$

Step 2. Plot the points on the coordinate plane.

Step 3. Observe the graph and draw conclusion.

From the graph, it can be observed that the costs of a member increased by $15 for 3 visits and the costs of nonmembers increased by $90 for 3 visits. 

Or we can say that the costs of a member increased by $5 per visit and the costs of nonmembers increased by $30 per visit.