Question
A ball is launched upward at 64 feet per second from a platform that is 36 feet high. The function that models this situation is h(t)=−16t2+64t+36 where t is the time in seconds and h(t) is the height of the ball. Find the maximum height the ball reaches and how long it will take to get there. (1 point) A) What key feature of the graph is the problem asking you to find? (answer choices: the vertex, y-intercept, x-intercept) B) The ball will reach a maximum height of feet in seconds.
Answers
GPT-4o mini
A) The problem is asking you to find the **vertex** of the parabola represented by the function \( h(t) = -16t^2 + 64t + 36 \). The vertex of a parabola in this context represents the maximum height reached by the ball.
B) To find the maximum height and the time it takes to reach that height, we can use the vertex formula for the quadratic function \( h(t) = at^2 + bt + c \). The time at which the maximum height occurs can be found using the formula:
\[
t = -\frac{b}{2a}
\]
In this case, \( a = -16 \) and \( b = 64 \):
\[
t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \text{ seconds}
\]
Next, we can substitute \( t = 2 \) back into the height function to find the maximum height:
\[
h(2) = -16(2^2) + 64(2) + 36
\]
\[
h(2) = -16(4) + 128 + 36
\]
\[
h(2) = -64 + 128 + 36
\]
\[
h(2) = 64 + 36 = 100 \text{ feet}
\]
Thus, the answers are:
**A)** The key feature of the graph is the **vertex**.
**B)** The ball will reach a maximum height of **100 feet** in **2 seconds**.
B) To find the maximum height and the time it takes to reach that height, we can use the vertex formula for the quadratic function \( h(t) = at^2 + bt + c \). The time at which the maximum height occurs can be found using the formula:
\[
t = -\frac{b}{2a}
\]
In this case, \( a = -16 \) and \( b = 64 \):
\[
t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \text{ seconds}
\]
Next, we can substitute \( t = 2 \) back into the height function to find the maximum height:
\[
h(2) = -16(2^2) + 64(2) + 36
\]
\[
h(2) = -16(4) + 128 + 36
\]
\[
h(2) = -64 + 128 + 36
\]
\[
h(2) = 64 + 36 = 100 \text{ feet}
\]
Thus, the answers are:
**A)** The key feature of the graph is the **vertex**.
**B)** The ball will reach a maximum height of **100 feet** in **2 seconds**.