Chapter 4.6, Problem 9CYU

To determine

To find:

The exact solution by using the given quadratic equation.

Answer to Problem 9CYU

The value of exact solution $t=0.78$ is reasonable.

Given:

The quadratic equation

  $h=-16t^2-64t+60=0$

Concept used:

The number is the square root of the quadratic formula.

  $b^2-4ac$

The discriminant formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The quadratic equation

  $h=-16t^2-64t+\mathrm{60.......................}\left(1\right)$

Substituting $0\text{for}h$ in the given function.

  $0=-16t^2-64t+60$

  $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{..................}\left(2\right)$

Comparing the quadratic equation $a{x}^2+bx+c=0$

  $a=-16,b=-64,c=60$

Putting the value of $a,b,c$ in equation (2)

  $\begin{array}{l}t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ t=\frac{-\left(-64\right)\pm \sqrt{{\left(-64\right)}^2-4\left(-16\right)\left(60\right)}}{2\left(-16\right)}\\ t=\frac{64\pm \sqrt{7936}}{-32}\\ t=\frac{64\pm 16\sqrt{31}}{-32}\\ t=-4.78\text{or}0.78\\ t=0.78\end{array}$

Therefore,

The value of exact solution $t=0.78$ is reasonable.

Explanation of Solution

Given:

The quadratic equation

  $h=-16t^2-64t+60=0$

Concept used:

The number is the square root of the quadratic formula.

  $b^2-4ac$

The discriminant formula

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The quadratic equation

  $h=-16t^2-64t+\mathrm{60.......................}\left(1\right)$

Substituting $0\text{for}h$ in the given function.

  $0=-16t^2-64t+60$

  $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\mathrm{..................}\left(2\right)$

Comparing the quadratic equation $a{x}^2+bx+c=0$

  $a=-16,b=-64,c=60$

Putting the value of $a,b,c$ in equation (2)

  $\begin{array}{l}t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ t=\frac{-\left(-64\right)\pm \sqrt{{\left(-64\right)}^2-4\left(-16\right)\left(60\right)}}{2\left(-16\right)}\\ t=\frac{64\pm \sqrt{7936}}{-32}\\ t=\frac{64\pm 16\sqrt{31}}{-32}\\ t=-4.78\text{or}0.78\\ t=0.78\end{array}$

Therefore,

The value of exact solution $t=0.78$ is reasonable.