Chapter 3.4, Problem 21PPS

To determine

To find: The number of raids in roller coaster, bumper car and water slide.

Answer to Problem 21PPS

There are 5 roller coaster, 1 bumper car and 4 water slide.

Explanation of Solution

Given information :

The wait for roller coaster is 1 hr.

The wait for the bumper car is 20 min.

The wait for the water slides is 15 min.

Calculation :

Consider x to be the number of raids in roller coaster, y to be the number of bumper car and z to the number of water slide.

The system of equation representing the situation is written as follows:

  $\begin{array}{l}x+y+z=10\mathrm{.........}\left(1\right)\hfill \\ x=y+z\mathrm{...........}\left(2\right)\hfill \\ 12x+4y+3z=76\mathrm{...........}\left(3\right)\hfill \end{array}$

Solve the system of equations as follows:

Substitute $x=y+z$ in first equation;

  $\begin{array}{l}x+y+z=10\hfill \\ x+x=10\hfill \\ 2x=10\hfill \\ x=5\hfill \end{array}$

Add equation two and three as follows:

  $\begin{array}{l}x\left(–12\right)\Rightarrow –12x+12y+12z=0\hfill \\ 12x+4y+3z=76\hfill \\ 16y+15z=76\hfill \end{array}$

Solve equation $16y+15z=76$ and second equation as follows:

  $\begin{array}{l}x\left(16\right)\Rightarrow –16x–16y–16z=0\hfill \\ 16y+15z=76\hfill \\ 16x–z=76\hfill \end{array}$

Substitute $x=5in16x–z=76$ as follows:

  $\begin{array}{l}16\left(5\right)–z=76\hfill \\ –z=76+80\hfill \\ x=4\hfill \end{array}$

Substitute $x=5$ and $z=4$ in equation two as follows:

  $\begin{array}{l} x=y+z\hfill \\ 5=y+4\hfill \\ 5–4=y\hfill \\ y=1\hfill \end{array}$

The solution set is $\left(5,1,4\right).$

Hence, there are 5 rollers coaster, 1 bumper car and 4 water slide.