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]
[tex]3\leq h\leq 67[/tex]
Step-by-step explanation:
In this problem we know that the equation is of the form
[tex]h(t) = -16t^2 + v_0t + h_0[/tex]
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
[tex]h(t) = -16t^2 + 64t + 3[/tex]
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
[tex]h(t) = -16t^2 + 64t + 3[/tex]
This is the equation of a vertical parabola open downward
The vertex represent a maximum
Convert the quadratic equation into vertex form
Factor -16
[tex]h(t) = -16(t^2 -4t) + 3[/tex]
Complete the square
[tex]h(t) = -16(t^2 -4t+4) + 3+64[/tex]
[tex]h(t) = -16(t^2 -4t+4) + 67[/tex]
Rewrite as perfect squares
[tex]h(t) = -16(t-2)^2) + 67[/tex]
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]
[tex]3\leq h\leq 67[/tex]
