To determine To find: a. The parametric equations that model the position of the ball as a function of time. Answer a. x = ( v o 1 2 ) t = 2 v o t 2 ; y = − 16 t ² + √ 2 v o 2 t + 3 Explanation Given: The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45 ° . Use g = 32 feet per secondÂ² as the acceleration due to gravity and ignore any air resistance. Formula used: x = ( v o cos θ ) t , y = − 1 2 g t ² + ( v o sin θ ) t + h Calculation: θ = 45 ° , g = 32 f t / s e c ² and h = 3 f e e t . a. x = ( v o cos θ ) t , y = − 1 2 g t ² + ( v o sin θ ) t + h x = ( v o cos 45 ) t x = ( v o 1 2 ) t = 2 v o t 2 h = 3 f e e t y = − 1 2 &*#x00A0; 32 t ² + ( v o sin 45 ) t + 3 y = − 16 t ² + √ 2 v o 2 t + 3 To determine To find: b. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet. Answer b. Maximum height = 139.1 f e e t Explanation Given: The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45 ° . Use g = 32 feet per secondÂ² as the acceleration due to gravity and ignore any air resistance. Formula used: x = ( v o cos θ ) t , y = − 1 2 g t ² + ( v o sin θ ) t + h Calculation: θ = 45 ° , g = 32 f t / s e c ² and h = 3 f e e t . b. Let v o = 90 miles/hr = 132 f t / s e c y = − 16 t ² + √ 2 v o 2 t + 3 y = − 16 t 2 + 132 2 2 t + 3 = − 16 t ² + 66 2 t + 3 Height is maximum when t = − b / 2 a t = − 66 2 2 &*#x00A0; − 16 ≈ 2.9168 sec y = − 16 ( 2.9168 ) 2 + 66 √ 2 ( 2.9168 ) + 3 ≈ 139.1 feet. To determine To find: c. How far is the ball from home plate at its maximum height? Give your answer in feet. Answer c. The ball is 272.25 feet from home plate at its maximum height. Explanation Given: The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45 ° . Use g = 32 feet per secondÂ² as the acceleration due to gravity and ignore any air resistance. Formula used: x = ( v o cos θ ) t , y = − 1 2 g t ² + ( v o sin θ ) t + h Calculation: θ = 45 ° , g = 32 f t / s e c ² and h = 3 f e e t . c. Horizontal distance x = ( v o cos θ ) t . From b height is maximum when t = 2.9168 sec. x = 132 2 √ 2 &*#x00A0; 2.9168 ≈ 272.25 feet. The ball is 272.25 feet from home plate at its maximum height. To determine To find: d. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall? Answer d. The ball will clear the Green Monster by 136.5 − 37 = 99.5 feet. Explanation Given: The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of 45 ° . Use g = 32 feet per secondÂ² as the acceleration due to gravity and ignore any air resistance. Formula used: x = ( v o cos θ ) t , y = − 1 2 g t ² + ( v o sin θ ) t + h Calculation: θ = 45 ° , g = 32 f t / s e c ² and h = 3 f e e t . d. The ball will reach the left field fence when x = 310 feet. x = ( v o cos θ ) t = 66 √ 2 t 310 66 2 = t ≈ 3.3213 sec Thus, it will take about 3.3213 sec to reach the left field fence. y = − 16 ( 3.3213 ) 2 + 66 2 ( 3.3213 ) + 3 ≈ 136.5 feet Field fence height = 37 feet, The ball will clear the Green Monster by 136.5 − 37 = 99.5 feet.

Precalculus Enhanced with Graphing Utilities

6th Edition

ISBN: 9780321795465

Author: Michael Sullivan, Michael III Sullivan

Publisher: PEARSON

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Textbook Question

Chapter 10.7, Problem 59AYU

The Green Monster The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45^{\circ}$ . Use $g = 32$ feet per second2 as the acceleration due to gravity and ignore any air resistance.

a. Find parametric equations that model the position of the ball as a function of time. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet.

b. How far is the ball from home plate at its maximum height? Give your answer in feet.

c. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?

Expert Solution

To determine

To find:

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

Answer to Problem 59AYU

a. $x = (v_{o} \frac{1}{\sqrt{2}}) t = \frac{\sqrt{2} v_{o} t}{2}$ ; $y = - 16 t ² + \sqrt 2 \frac{v_{o}}{2} t + 3$

Explanation of Solution

Given:

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45 °$ . Use $g = 32$ feet per secondÂ² as the acceleration due to gravity and ignore any air resistance.

Formula used:

$x = (v_{o} cos θ) t$ , $y = - \frac{1}{2} g t ² + (v_{o} sin θ) t + h$

Calculation:

$θ = 45 °$ , $g = 32 f t / s e c ²$ and $h = 3 f e e t$ .

a. $x = (v_{o} cos θ) t$ , $y = - \frac{1}{2} g t ² + (v_{o} sin θ) t + h$

$x = (v_{o} cos 45) t$

$x = (v_{o} \frac{1}{\sqrt{2}}) t = \frac{\sqrt{2} v_{o} t}{2}$

$h = 3 f e e t$

$y = - \frac{1}{2} &*#x00A0; 32 t ² + (v_{o} \sin 45) t + 3$

$y = - 16 t ² + \sqrt 2 \frac{v_{o}}{2} t + 3$

Expert Solution

To determine

To find:

b. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet.

Answer to Problem 59AYU

b. Maximum height $= 139.1 f e e t$

Explanation of Solution

Given:

Formula used:

$x = (v_{o} cos θ) t$ , $y = - \frac{1}{2} g t ² + (v_{o} sin θ) t + h$

Calculation:

$θ = 45 °$ , $g = 32 f t / s e c ²$ and $h = 3 f e e t$ .

b. Let $v_{o} = 90$ miles/hr $= 132 f t / s e c$

$y = - 16 t ² + \sqrt 2 \frac{v_{o}}{2} t + 3$

$y = - 16 t^{2} + \frac{132}{2} \sqrt{2} t + 3 = - 16 t ² + 66 \sqrt{2} t + 3$

Height is maximum when $t = - b / 2 a$

$t = - \frac{66 \sqrt{2}}{2 &*#x00A0; - 16} \approx 2.9168$ sec

$y = - 16 {(2.9168)}^{2} + 66 \sqrt 2 (2.9168) + 3 \approx 139.1$ feet.

Expert Solution

To determine

To find:

c. How far is the ball from home plate at its maximum height? Give your answer in feet.

Answer to Problem 59AYU

c. The ball is $272.25$ feet from home plate at its maximum height.

Explanation of Solution

Given:

Formula used:

$x = (v_{o} cos θ) t$ , $y = - \frac{1}{2} g t ² + (v_{o} sin θ) t + h$

Calculation:

$θ = 45 °$ , $g = 32 f t / s e c ²$ and $h = 3 f e e t$ .

c. Horizontal distance $x = (v_{o} cos θ) t$ .

From b height is maximum when $t = 2.9168$ sec.

$x = \frac{132}{2} \sqrt 2 &*#x00A0; 2.9168 \approx 272.25$ feet.

The ball is $272.25$ feet from home plate at its maximum height.

Expert Solution

To determine

To find:

d. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?

Answer to Problem 59AYU

d. The ball will clear the Green Monster by $136.5 - 37 = 99.5$ feet.

Explanation of Solution

Given:

Formula used:

$x = (v_{o} cos θ) t$ , $y = - \frac{1}{2} g t ² + (v_{o} sin θ) t + h$

Calculation:

$θ = 45 °$ , $g = 32 f t / s e c ²$ and $h = 3 f e e t$ .

d. The ball will reach the left field fence when $x = 310$ feet.

$x = (v_{o} cos θ) t = 66 \sqrt 2 t$

$\frac{310}{66 \sqrt{2}} = t \approx 3.3213$ sec

Thus, it will take about $3.3213$ sec to reach the left field fence.

$y = - 16 {(3.3213)}^{2} + 66 \sqrt{2} (3.3213) + 3 \approx 136.5$ feet

Field fence height $= 37$ feet,

The ball will clear the Green Monster by $136.5 - 37 = 99.5$ feet.

Chapter 10.7, Problem 58AYU

Chapter 10.7, Problem 60AYU

Chapter 10 Solutions

Precalculus Enhanced with Graphing Utilities

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