Chapter 8.7, Problem 42PS

Solution

To solve: The system of equation by using both elimination and substitution method.

The ball hits the ceiling at the point $\left(0,36\right)$ .

Given information: The equation for the dome-shaped ceiling is given by $x^2+y^2+60y-3456=0$ and the equation for the shot that the tennis player hits is $x^2+y=36$ .

Calculation:

Now take the two given equation and solve using substitution method,

  $\begin{array}{c}{x}^2+x^2+y^2+60y-3456=0\mathrm{......}\left(1\right)\\ x^2+y=36\mathrm{......}\left(2\right)\end{array}$

Rearrange the second equation,

  $\begin{array}{c}{x}^2+y=36\\ x^2=36-y\end{array}$

Substitute $x^2=36-y$ in the first equation,

  $\begin{array}{c}36-y+y^2+60y-3456=0\\ y^2+59y-3420=0\end{array}$

Solving the above equation to get, $y_1=-95\text{ and }{y}_2=36$ . Length cannot have negative values. So the value $y_2=36$ alone is taken.

Now substitute $y_2=36$ in the second equation,

  $\begin{array}{c}{x}^2=36-y\\ =36-36\\ x^2=0\end{array}$

Using the elimination method, multiply the second equation with $\text{-1}$

  $\begin{array}{c}{y}^2+59y-3456=-36\\ y^2+59y-3420=0\end{array}$

Thus after solving, the same equation is obtained.

Therefore the ball hits the ceiling is at the point $\left(0,36\right)$ .