Question
A golf ball is hit from the ground at an upward velocity of 36 meters per second. This is modeled by the equation, h=−4.9t2+36t
. Using this function, determine whether the following scenario is true or false: The golf ball is hit from the ground initially. After traveling for roughly 3.7 seconds, it will reach a maximum height of 66 meters, where it will then begin its descent and land back on the ground after 9 seconds.
Type 1 for true.
Type 2 for false.
(1 point)
. Using this function, determine whether the following scenario is true or false: The golf ball is hit from the ground initially. After traveling for roughly 3.7 seconds, it will reach a maximum height of 66 meters, where it will then begin its descent and land back on the ground after 9 seconds.
Type 1 for true.
Type 2 for false.
(1 point)
Answers
GPT-4o mini
To determine the validity of the scenario presented, we'll analyze the function given and calculate the maximum height and the time it takes for the ball to hit the ground.
The height \( h \) of the golf ball at time \( t \) is given by:
\[
h = -4.9t^2 + 36t
\]
1. **Maximum Height:**
The maximum height of a quadratic function \( h(t) = at^2 + bt + c \) occurs at \( t = -\frac{b}{2a} \). In this case, \( a = -4.9 \) and \( b = 36 \).
\[
t = -\frac{36}{2 \times -4.9} = \frac{36}{9.8} \approx 3.673 \text{ seconds}
\]
Now we will calculate \( h(3.673) \):
\[
h(3.673) = -4.9(3.673)^2 + 36(3.673)
\]
\[
h(3.673) = -4.9(13.532) + 132.168 \approx -66.743 + 132.168 \approx 65.425 \text{ meters}
\]
The maximum height is approximately 65.425 meters, which is approximately 66 meters. This part of the scenario holds true.
2. **Time of Flight:**
To find the total time of flight until the ball lands back on the ground, we set \( h = 0 \):
\[
-4.9t^2 + 36t = 0
\]
Factor out \( t \):
\[
t(-4.9t + 36) = 0
\]
This gives us two solutions: \( t = 0 \) (when the ball is hit) or:
\[
-4.9t + 36 = 0 \implies 4.9t = 36 \implies t \approx \frac{36}{4.9} \approx 7.347 \text{ seconds}
\]
Thus, the golf ball lands back on the ground after approximately 7.347 seconds.
Based on the calculations:
- The ball reaches a maximum height close to 66 meters after approximately 3.7 seconds.
- The ball lands back on the ground after approximately 7.347 seconds, not after 9 seconds.
Therefore, the statement in the scenario is **false**.
The answer is: **2 (false)**.
The height \( h \) of the golf ball at time \( t \) is given by:
\[
h = -4.9t^2 + 36t
\]
1. **Maximum Height:**
The maximum height of a quadratic function \( h(t) = at^2 + bt + c \) occurs at \( t = -\frac{b}{2a} \). In this case, \( a = -4.9 \) and \( b = 36 \).
\[
t = -\frac{36}{2 \times -4.9} = \frac{36}{9.8} \approx 3.673 \text{ seconds}
\]
Now we will calculate \( h(3.673) \):
\[
h(3.673) = -4.9(3.673)^2 + 36(3.673)
\]
\[
h(3.673) = -4.9(13.532) + 132.168 \approx -66.743 + 132.168 \approx 65.425 \text{ meters}
\]
The maximum height is approximately 65.425 meters, which is approximately 66 meters. This part of the scenario holds true.
2. **Time of Flight:**
To find the total time of flight until the ball lands back on the ground, we set \( h = 0 \):
\[
-4.9t^2 + 36t = 0
\]
Factor out \( t \):
\[
t(-4.9t + 36) = 0
\]
This gives us two solutions: \( t = 0 \) (when the ball is hit) or:
\[
-4.9t + 36 = 0 \implies 4.9t = 36 \implies t \approx \frac{36}{4.9} \approx 7.347 \text{ seconds}
\]
Thus, the golf ball lands back on the ground after approximately 7.347 seconds.
Based on the calculations:
- The ball reaches a maximum height close to 66 meters after approximately 3.7 seconds.
- The ball lands back on the ground after approximately 7.347 seconds, not after 9 seconds.
Therefore, the statement in the scenario is **false**.
The answer is: **2 (false)**.