Chapter 9.1, Problem 63PPS

b.

To determine

To calculate the height from which the ball was hit.

Answer to Problem 63PPS

The height from which the ball was hit is the ground level, or, $h=0$ meters.

Explanation of Solution

Given information :

The function provided is $h=-4.9t^2+31.3t$ .

Formula used :

To compute the height from which the ball was hit, find the y-intercept of the given equation.

y-intercept is computed by substituting the value of t in the equation by 0

Calculation :

  $=-4.9{\left(0\right)}^2+31.3\left(0\right)$

Simplifying

  $=0$

Hence the point of y-intercept is $\left(0,0\right)$ .

Thus the ball was hit from the ground level at $h=0$ meters.

c.

To determine

To calculate maximum height reached by the ball.

Answer to Problem 63PPS

The maximum height reached by the ball is $h=49.984$ meters.

Explanation of Solution

Given information :

The function provided is $h=-4.9t^2+31.3t$ .

Formula used :

The maximum height attained by the ball can be calculated by finding the vertex of the equation.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ . Here, $a=-4.9$ and $b=31.3$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively.

Axis of symmetry for the given function is

  $x=-\frac{b}{2\times a}$

Put this value of x in the main function to get h this will be maximum height.

Calculation :

Axis of symmetry for the given function is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{31.3}{2\times \left(-4.9\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{31.3}{9.8}$

Simplifying this

  $x=3.194$ is the equation of axis of symmetry.

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y . These two coordinates of x and y would be the point where the vertex is.

  $=-4.9{\left(3.194\right)}^2+31.3\left(3.194\right)$

Putting the value of $x=3.194$ in the original function.

  $=-49.981+99.965$

Simplifying the expression.

  $=49.984$

Thus the maximum height reached is $h=49.984$ meters.

d.

To determine

To calculate the time taken by the ball to hit the ground.

Answer to Problem 63PPS

The time taken by the ball to hit the ground is $t=6.3878$ seconds.

Explanation of Solution

Given information :

The function provided is $h=-4.9t^2+31.3t$ .

Formula used :

To calculate the total time taken, use the formula to find roots of a quadratic equation. This can be found by using $t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ . Here ‘a’, ‘b’ and ‘c’ are the co-efficient of the given equation.

Here, $a=-4.9$ , $b=31.3$ and $c=0$ .

Calculation :

  $t=\frac{-31.3\pm \sqrt{{31.3}^2-4\times -4.9\times 0}}{2\times \left(-4.9\right)}$

Putting the values in the formula.

  $=\frac{-31.3\pm \sqrt{{31.3}^2}}{-9.8}$

Simplifying

  $=\frac{-31.3\pm 31.3}{-9.8}$

Removing square root.

  $=\frac{-31.3+31.3}{-9.8}$

One of the roots simplifications.

  $=0$

This is the first root.

  $=\frac{-31.3-31.3}{-9.8}$

Second root’s simplification.

  $=\frac{-62.6}{-9.8}$

Dividing the fraction.

  $t=6.3878$

Thus, the ball hits the ground at $t=6.3878$ seconds.

e.

To determine

To mention a reasonable domain and range for the given equation.

Answer to Problem 63PPS

A reasonable domain for this function would be $\left\{0\le t\le 6.3878\right\}$ and a reasonable range would be $\left\{h|h\le 49.984\right\}$ , since h and t cannot take negative values.

Explanation of Solution

Given information :

The function provided is $h=-4.9t^2+31.3t$ .

Since the given function is a quadratic one, the graph is that of a parabola. In case of a parabola, the domain is all values with t can take. This is case, it is evident that the ball would not have a negative time and neither would the value of h be negative. Thus, the domain is $\left\{0\le t\le 6.3878\right\}$ and the range is $\left\{0\le h\le 49.984\right\}$ .

Had this curve not represented any ball moving trajectory, the domain would have been all real numbers while the range would have been $\left\{h|h\le 49.984\right\}$ .