Chapter 10, Problem 43RE

To determine

To find:

a. Find parametric equations that model the position of the ball as a function of time.

Answer to Problem 43RE

a. $x=\left(80\text{cos}35°\right)t$, $y=-16t^2+(80\sin 35°)t+6$

Explanation of Solution

Given:

Drew Brees throws a football with an initial speed of 80 feet per second at an angle of $35°$ to the horizontal. The ball leaves Brees’ hand at a height of 6 feet.

Formula used:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

Calculation:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

a. $v_0=80$, $\theta =35°$

$x=\left(80\text{cos}35\right)t$

$y=-\frac{1}{2}*32t^2+(80\sin 35)t+6$

$y=-16t^2+(80\sin 35)t+6$

To determine

To find:

b. How long is the ball in the air?

Answer to Problem 43RE

b. $t=2.99$ seconds

Explanation of Solution

Given:

Drew Brees throws a football with an initial speed of 80 feet per second at an angle of $35°$ to the horizontal. The ball leaves Brees’ hand at a height of 6 feet.

Formula used:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

Calculation:

b. The ball is in air until $y=0$.

$y=-16t^2+(80\sin 35°)t+6$

$0=-16t^2+45.89t+6$

$t=\frac{4589-\sqrt{24898921}}{3200} \text{or} t=\frac{\sqrt{24898921}+4589}{3200}$

$t\approx -0.13 \text{or} 2.99$

The ball is in the air for about $c$.

To determine

To find:

c. When is the ball at its maximum height? Determine the maximum height of the ball.

Answer to Problem 43RE

c. Maximum height of the ball: $38.9$ feet.

Explanation of Solution

Given:

Drew Brees throws a football with an initial speed of 80 feet per second at an angle of $35°$ to the horizontal. The ball leaves Brees’ hand at a height of 6 feet.

Formula used:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

Calculation:

c. The maximum height occurs at the vertex of the quadratic function.

$t=-\frac{b}{2a}=-\frac{80\text{sin}35}{2\left(-16\right)}\approx 1.43$ seconds

$y=-16{\left(1.43\right)}^2+80\text{sin}35°\left(1.43\right)+6\approx 38.9$ feet

Maximum height of the ball is $38.9$ feet.

To determine

To find:

d. Determine the horizontal distance that the ball travels.

Answer to Problem 43RE

d. Horizontal distance: 196 feet.

Explanation of Solution

Given:

Drew Brees throws a football with an initial speed of 80 feet per second at an angle of $35°$ to the horizontal. The ball leaves Brees’ hand at a height of 6 feet.

Formula used:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

Calculation:

d. To determine the horizontal distance that the ball travels.

$x=\left(80\text{cos}35\right)t$

$x=\left(80\text{cos}35\right)\left(2.99\right)\approx 196$ feet

To determine

To find:

e. Using a graphing utility, simultaneously graph the equations found in part a.

Answer to Problem 43RE

e. Graph is plotted.

Explanation of Solution

Given:

Drew Brees throws a football with an initial speed of 80 feet per second at an angle of $35°$ to the horizontal. The ball leaves Brees’ hand at a height of 6 feet.

Formula used:

$x=v_0\text{cos}\theta t;y=-\frac{1}{2}g{t}^2+\left(v_o\text{sin}\theta \right)t+h$

Calculation:

e. The graph of equation found in part a.