Question
The height h of a fireball launched from a Roman candle with an initial velocity of 96 feet per second is given by the equation hequalsminus16tsquaredplus96t where t is time in seconds after launch. Use the graph of this function to answer the questions.
0
2
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8
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100
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300
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Feet
t
h
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300
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h
Answers
GPT 3.5
To answer questions about the height of the fireball at different times, we need to plug in the given values of t (time) into the equation h = -16t^2 + 96t.
When t = 0:
h = -16(0)^2 + 96(0)
h = 0 feet
When t = 2:
h = -16(2)^2 + 96(2)
h = -16(4) + 192
h = -64 + 192
h = 128 feet
When t = 4:
h = -16(4)^2 + 96(4)
h = -16(16) + 384
h = -256 + 384
h = 128 feet
When t = 6:
h = -16(6)^2 + 96(6)
h = -16(36) + 576
h = -576 + 576
h = 0 feet
When t = 8:
h = -16(8)^2 + 96(8)
h = -16(64) + 768
h = -1024 + 768
h = -256 feet
Therefore, the height of the fireball at different times after launch is as follows:
- At time t = 0 seconds, the height is 0 feet.
- At time t = 2 seconds, the height is 128 feet.
- At time t = 4 seconds, the height is 128 feet.
- At time t = 6 seconds, the height is 0 feet.
- At time t = 8 seconds, the height is -256 feet.
When t = 0:
h = -16(0)^2 + 96(0)
h = 0 feet
When t = 2:
h = -16(2)^2 + 96(2)
h = -16(4) + 192
h = -64 + 192
h = 128 feet
When t = 4:
h = -16(4)^2 + 96(4)
h = -16(16) + 384
h = -256 + 384
h = 128 feet
When t = 6:
h = -16(6)^2 + 96(6)
h = -16(36) + 576
h = -576 + 576
h = 0 feet
When t = 8:
h = -16(8)^2 + 96(8)
h = -16(64) + 768
h = -1024 + 768
h = -256 feet
Therefore, the height of the fireball at different times after launch is as follows:
- At time t = 0 seconds, the height is 0 feet.
- At time t = 2 seconds, the height is 128 feet.
- At time t = 4 seconds, the height is 128 feet.
- At time t = 6 seconds, the height is 0 feet.
- At time t = 8 seconds, the height is -256 feet.
a. Estimate the maximum height of the fireball.
The maximum height of the fireball is nothing ft.
The maximum height of the fireball is nothing ft.
GPT 3.5
To estimate the maximum height of the fireball, we need to find the vertex of the quadratic equation h = -16t^2 + 96t. The vertex of a parabola given by the equation y = ax^2 + bx + c is located at x = -b / (2a).
In this case, a = -16 and b = 96. Therefore, the time at which the maximum height occurs is t = -96 / (2*(-16)) = 3 seconds.
Now, we can find the maximum height by plugging this value of t back into the equation:
h = -16(3)^2 + 96(3)
h = -16(9) + 288
h = -144 + 288
h = 144 feet
Therefore, the estimated maximum height of the fireball is 144 feet.
In this case, a = -16 and b = 96. Therefore, the time at which the maximum height occurs is t = -96 / (2*(-16)) = 3 seconds.
Now, we can find the maximum height by plugging this value of t back into the equation:
h = -16(3)^2 + 96(3)
h = -16(9) + 288
h = -144 + 288
h = 144 feet
Therefore, the estimated maximum height of the fireball is 144 feet.
a. Estimate the maximum height of the fireball.
The maximum height of the fireball is 144 ft.
Part 2
b. Estimate the time when the fireball is at its maximum height.
The time when the fireball is at its maximum height is tequals
enter your response here sec.
The maximum height of the fireball is 144 ft.
Part 2
b. Estimate the time when the fireball is at its maximum height.
The time when the fireball is at its maximum height is tequals
enter your response here sec.
GPT 3.5
The time when the fireball is at its maximum height is 3 seconds.
a. Estimate the maximum height of the fireball.
The maximum height of the fireball is 144 ft.
Part 2
b. Estimate the time when the fireball is at its maximum height.
The time when the fireball is at its maximum height is tequals
3 sec.
Part 3
c. Estimate the time when the fireball returns to the ground.
The time when the fireball returns to the ground is tequals
enter your response here sec.
The maximum height of the fireball is 144 ft.
Part 2
b. Estimate the time when the fireball is at its maximum height.
The time when the fireball is at its maximum height is tequals
3 sec.
Part 3
c. Estimate the time when the fireball returns to the ground.
The time when the fireball returns to the ground is tequals
enter your response here sec.
GPT 3.5
The time when the fireball returns to the ground can be found by setting the height equation h = -16t^2 + 96t to 0 and solving for t. This represents the time when the fireball hits the ground.
-16t^2 + 96t = 0
-16t(t - 6) = 0
t = 0 or t = 6
Since t = 0 corresponds to the moment of launch, the fireball returns to the ground after 6 seconds.
Therefore, the time when the fireball returns to the ground is 6 seconds.
-16t^2 + 96t = 0
-16t(t - 6) = 0
t = 0 or t = 6
Since t = 0 corresponds to the moment of launch, the fireball returns to the ground after 6 seconds.
Therefore, the time when the fireball returns to the ground is 6 seconds.