Question
The height h of a fireball launched from a Roman candle with an initial velocity of 112 feet per second is given by the equation hequalsminus16tsquaredplus112t where t is time in seconds after launch. Use the graph of this function to answer the questions.
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Answers
GPT 3.5
Based on the equation h = -16t^2 + 112t, we can calculate the height of the fireball at different time intervals:
- At t = 0 seconds: h = -16(0)^2 + 112(0) = 0
- At t = 2 seconds: h = -16(2)^2 + 112(2) = 32 feet
- At t = 4 seconds: h = -16(4)^2 + 112(4) = 128 feet
- At t = 6 seconds: h = -16(6)^2 + 112(6) = 144 feet
- At t = 8 seconds: h = -16(8)^2 + 112(8) = 128 feet
By looking at the values calculated above and the graph of the function, we can see that the maximum height of the fireball occurs at t = 6 seconds, where h = 144 feet. At t = 0 and t = 8 seconds, the fireball is at t = 0 feet.
The height of the fireball decreases after reaching the maximum height, and it hits the ground again at t = 8 seconds when it reaches 0 feet.
- At t = 0 seconds: h = -16(0)^2 + 112(0) = 0
- At t = 2 seconds: h = -16(2)^2 + 112(2) = 32 feet
- At t = 4 seconds: h = -16(4)^2 + 112(4) = 128 feet
- At t = 6 seconds: h = -16(6)^2 + 112(6) = 144 feet
- At t = 8 seconds: h = -16(8)^2 + 112(8) = 128 feet
By looking at the values calculated above and the graph of the function, we can see that the maximum height of the fireball occurs at t = 6 seconds, where h = 144 feet. At t = 0 and t = 8 seconds, the fireball is at t = 0 feet.
The height of the fireball decreases after reaching the maximum height, and it hits the ground again at t = 8 seconds when it reaches 0 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
The maximum height of the fireball is estimated to be 144 feet.
a. Estimate the maximum height of the fireball.
The maximum height of the fireball is 196 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 196 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
Based on the information provided earlier and the estimated maximum height of 196 feet, we can recalculate the time at which the fireball reaches its maximum height using the equation h = -16t^2 + 112t.
To find the time when the fireball is at its maximum height, we need to calculate the vertex of the parabolic function represented by the equation h = -16t^2 + 112t. The time at the vertex point of a parabola is given by the formula: t = -b/(2a), where a is the coefficient of the t^2 term and b is the coefficient of the t term in the equation.
In this case, a = -16 and b = 112. Substituting these values into the formula, we get:
t = -112/(2*(-16))
t = -112/(-32)
t ≈ 3.5 seconds
Therefore, the estimated time when the fireball is at its maximum height is around 3.5 seconds.
To find the time when the fireball is at its maximum height, we need to calculate the vertex of the parabolic function represented by the equation h = -16t^2 + 112t. The time at the vertex point of a parabola is given by the formula: t = -b/(2a), where a is the coefficient of the t^2 term and b is the coefficient of the t term in the equation.
In this case, a = -16 and b = 112. Substituting these values into the formula, we get:
t = -112/(2*(-16))
t = -112/(-32)
t ≈ 3.5 seconds
Therefore, the estimated time when the fireball is at its maximum height is around 3.5 seconds.
a. Estimate the maximum height of the fireball.
The maximum height of the fireball is 196 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
3.5 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 196 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
3.5 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
To find the time when the fireball returns to the ground, we can set h = 0 in the equation h = -16t^2 + 112t and solve for t:
0 = -16t^2 + 112t
16t^2 = 112t
16t = 112
t = 112 / 16
t = 7
Therefore, the time when the fireball returns to the ground is estimated to be 7 seconds.
0 = -16t^2 + 112t
16t^2 = 112t
16t = 112
t = 112 / 16
t = 7
Therefore, the time when the fireball returns to the ground is estimated to be 7 seconds.
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