A toy rocket is shot vertically into the air from a launching pad 66 feet above the ground with an initial velocity of 152152 feet per second. the height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h equals negative 16 t squared plus 152 t plus 6h=−16t2 152t 6. how long will it take the rocket to reach its maximum height? what is the maximum height?

Question

07-10-2023
Physics

High School

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A toy rocket is shot vertically into the air from a launching pad 66 feet above the ground with an initial velocity of 152152 feet per second. the height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h equals negative 16 t squared plus 152 t plus 6h=−16t2 152t 6. how long will it take the rocket to reach its maximum height? what is the maximum height?

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adarshscorium adarshscorium · Answer 1 · 2024-04-09T19:44:29-04:00

The time to reach maximum height is 4.75 seconds using the vertex formula for the quadratic function representing the rocket's height. The maximum height can then be calculated by substituting the time back into the function.

The question involves finding the time it takes for a toy rocket to reach its maximum height and determining what that height is, using the given quadratic function h(t) = -16t2 + 152t + 66. The maximum height of a projectile is reached when its vertical velocity component is zero.

This occurs at the vertex of the parabolic trajectory represented by the quadratic function, which can be calculated using the vertex formula t = -b/(2a), where a and b are the coefficients from the quadratic equation.

To calculate the maximum height, we substitute this time value back into the height function. For the given function h(t) = -16t2 + 152t + 66, the time to reach the maximum height is t = -152/(2(-16)) = 4.75 seconds. The maximum height reached by the rocket is h(4.75) = -16(4.75)2 + 152(4.75) + 66, which can be calculated to give the maximum height value.