Chapter 3.5, Problem 21PPS

To determine

To calculate: The number of roller coasters, bumper cars and water slides he ride.

Answer to Problem 21PPS

He rode 5roller coasters, 1 bumper car and 4 water slides.

Explanation of Solution

Given information:

The waiting period for roller coaster is 1 hour, for bumper car is 20 minutes and for water slide is 15 minutes. He rode 10 rides. Sum of number of times he bumper and water slide equals to number of times he rode roller coaster. He waited for a total of 6 hours and 20 minutes in each for rides.

Calculation:

It is provided that the waiting period for roller coaster is 1 hour, for bumper car is 20 minutes and for water slide is 15 minutes. He rode 10 rides. Sum of number of times he bumper and water slide equals to number of times he rode roller coaster. He waited for a total of 6 hours and 20 minutes in each for rides.

Let number of times he rode roller coaster isx , bumper car be y and water slide be z .

He rode 10 rides, so $x+y+z=10$ .

The waiting period for roller coaster is 1 hour, for bumper car is 20 minutes and for water slide is 15 minutes.He waited for a total of 6 hours and 20 minutes in each for rides.

So,

  $\begin{array}{c}60x+20y+15z=380\\ 12x+4y+3z=76\end{array}$

Sum of number of times he bumper and water slide equals to number of times he rode roller coaster.

So,

  $\begin{array}{c}y+z=x\\ -x+y+z=0\end{array}$

Consider the system of equations

  $\begin{array}{c}x+y+z=10\\ 12x+4y+3z=76\\ -x+y+z=0\end{array}$

Take the first and the third equation,

  $\begin{array}{c}x+y+z=10\\ -x+y+z=0\end{array}$

Use the method of elimination to solve the system above.

Subtract both the equations,

  $\begin{array}{ccccccc}x& +& y& +& z& =& 10\\ -x& +& y& +& z& =& 0\\ -& & & & & & \\ 2x& & & & & =& 10\end{array}$

Therefore,

  $\begin{array}{c}2x=10\\ x=5\end{array}$

Now substitute $x=5$ in the equations $x+y+z=10\text{ and }12x+4y+3z=76$ ,

  $\begin{array}{c}5+y+z=10\\ 60+4y+3z=76\end{array}$

Simplify the above system,

  $\begin{array}{c}y+z=5\\ 4y+3z=16\end{array}$

Use substitution to solve the above system.

From first equation,

  $y=5-z$

Now substitute $y=5-z$ and $4y+3z=16$ .

  $\begin{array}{c}4\left(5-z\right)+3z=16\\ 20-4z+3z=16\\ z=4\end{array}$

Now, substitute $z=4$ in $y=5-z$ .

  $\begin{array}{c}y=5-4\\ y=1\end{array}$

Therefore, solution is the coordinate pair, $\left(5,1,4\right)$ .

Thus, he rode 5 roller coasters, 1 bumper car and 4 water slides.