Let's break down each of the parts step by step.
### Part A: Find [tex]\( f(0) \)[/tex].
To find [tex]\( f(0) \)[/tex], substitute [tex]\( t = 0 \)[/tex] into the equation:
[tex]\[
f(t) = -16t^2 + 40t + 3.6
\][/tex]
[tex]\[
f(0) = -16(0)^2 + 40(0) + 3.6 = 3.6
\][/tex]
So,
[tex]\[
f(0) = 3.6 \text{ feet}
\][/tex]
### Part B: Find [tex]\( f(2) \)[/tex].
To find [tex]\( f(2) \)[/tex], substitute [tex]\( t = 2 \)[/tex] into the equation:
[tex]\[
f(t) = -16t^2 + 40t + 3.6
\][/tex]
[tex]\[
f(2) = -16(2)^2 + 40(2) + 3.6 = -64 + 80 + 3.6 = 19.6
\][/tex]
So,
[tex]\[
f(2) = 19.6 \text{ feet}
\][/tex]
### Part C: Find the time it takes for the golf ball to reach its maximum height.
The time at which a quadratic function [tex]\( at^2 + bt + c \)[/tex] reaches its maximum or minimum is given by [tex]\( t = -\frac{b}{2a} \)[/tex].
Here, [tex]\( a = -16 \)[/tex] and [tex]\( b = 40 \)[/tex].
[tex]\[
t = -\frac{40}{2 \cdot -16} = \frac{40}{32} = 1.25
\][/tex]
So, the golf ball reaches its maximum height at
[tex]\[
t = 1.25 \text{ seconds}
\][/tex]
### Part D: What is the maximum height?
To find the maximum height, substitute [tex]\( t = 1.25 \)[/tex] into the height function:
[tex]\[
f(t) = -16t^2 + 40t + 3.6
\][/tex]
[tex]\[
f(1.25) = -16(1.25)^2 + 40(1.25) + 3.6 = -16(1.5625) + 50 + 3.6 = 28.6
\][/tex]
So, the maximum height is
[tex]\[
28.6 \text{ feet}
\][/tex]
### Part E: When will the golf ball hit the ground?
The golf ball hits the ground when [tex]\( f(t) = 0 \)[/tex]:
[tex]\[
-16t^2 + 40t + 3.6 = 0
\][/tex]
Solving this quadratic equation, we get a positive root of
[tex]\[
t = 2.587
\][/tex]
So, the golf ball hits the ground at
[tex]\[
t = 2.587 \text{ seconds}
\][/tex]
### Part F: When will the golf ball be 15 feet above the ground?
To find when the golf ball is 15 feet above the ground, solve [tex]\( f(t) = 15 \)[/tex]:
[tex]\[
-16t^2 + 40t + 3.6 = 15
\][/tex]
Simplifying gives:
[tex]\[
-16t^2 + 40t - 11.4 = 0
\][/tex]
Solving this quadratic equation, the positive solutions are
[tex]\[
t = 0.328 \text{ seconds}, \quad t = 2.172 \text{ seconds}
\][/tex]
So, the times when the golf ball is 15 feet above the ground are
[tex]\[
0.328 \text{ seconds}, 2.172 \text{ seconds}
\][/tex]
By combining all the results, we get:
A) [tex]\( f(0) = 3.6 \text{ feet} \)[/tex]
B) [tex]\( f(2) = 19.6 \text{ feet} \)[/tex]
C) Time to maximum height: [tex]\( 1.25 \text{ seconds} \)[/tex]
D) Maximum height: [tex]\( 28.6 \text{ feet} \)[/tex]
E) Time when the golf ball hits the ground: [tex]\( 2.587 \text{ seconds} \)[/tex]
F) Times when the golf ball is 15 feet above the ground: [tex]\( 0.328 \text{ seconds}, 2.172 \text{ seconds} \)[/tex]
