Question

During halftime of a football game, a slingshot launcher’s T-shirts at the crowd. A T-shirt is launch from a height of 3 feet with an initial upward velocity of 64 ft./s. The T-shirt is caught at 45 feet above the field how long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function of models the height of the T-shirt over time

Answer 1

Answer:

Part A) The time it will take to the shirt to reach its maximum height is 2 seconds

Part B) The maximum height is 67 feet

Part C) The range is the interval [3,67]

$3\leq h\leq 67$

Step-by-step explanation:

In this problem we know that the equation is of the form

$h(t) = -16t^2 + v_0t + h_0$

where

-16 is the gravitational constant,

v₀ is the initial velocity in feet per second

h₀ is the initial height in feet

substitute the given values

$h(t) = -16t^2 + 64t + 3$

Part A) How long will it take the T-shirt to reach its maximum height?

we know that

The x-coordinate of the vertex is the time that it will take to the T-shirt to reach its maximum height

we have

$h(t) = -16t^2 + 64t + 3$

This is the equation of a vertical parabola open downward

The vertex represent a maximum

Convert the quadratic equation into vertex form

Factor -16

$h(t) = -16(t^2 -4t) + 3$

Complete the square

$h(t) = -16(t^2 -4t+4) + 3+64$

$h(t) = -16(t^2 -4t+4) + 67$

Rewrite as perfect squares

$h(t) = -16(t-2)^2) + 67$

The vertex is the point (2,67)

therefore

The time it will take to the shirt to reach its maximum height is 2 seconds

Part B) What is the maximum height?

we know that

The maximum height is equal to the y-coordinate of the vertex of the quadratic equation

The vertex is the point (2,67)

therefore

The maximum height is 67 feet

Part C) What is the range of the function of models the height of the T-shirt over time

we know that

The initial height is 3 feet, the maximum height is 67 feet and the T-shirt is caught at 45 feet above the field

so

The range is the interval [3,67]

$3\leq h\leq 67$