A toy rocket is shot vertically into the air from a launching pad 4 feet above the ground with an initial velocity of 88 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(t)=-16t^2+88t+4 How long will it take the rocket to reach its maximum height? What is the maximum height?
To find the time it takes for the rocket to reach its maximum height, we need to determine the time at which the vertex of the parabolic function occurs. The function h(t) is in the form of a quadratic equation, h(t) = at^2 + bt + c, where a = -16, b = 88, and c = 4.
The time at which the vertex occurs can be found using the formula t = -b/(2a). Plugging in the values, we have t = -88/(2*-16) = 11/4 seconds.
To find the maximum height, we substitute the value of t = 11/4 into the function h(t):
h(11/4) = -16(11/4)^2 + 88(11/4) + 4
= -484/16 + 968/4 + 4
= 242/8 + 968/4 + 4
= 124 + 968/4
= 124 + 242
= 366 feet.
Therefore, the rocket will reach its maximum height in 11/4 seconds, and the maximum height is 366 feet.
To find the time it takes for the rocket to reach its maximum height, we can determine the vertex of the parabolic function h(t)=-16t^2+88t+4.
The vertex of a parabola is given by the formula t = -b/2a, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -16 and b = 88. Plugging these values into the formula, we have:
t = -88 / (2 * -16)
t = -88 / -32
t = 2.75
Therefore, it will take the rocket approximately 2.75 seconds to reach its maximum height.
To find the maximum height, we substitute this time value back into the equation h(t) to get the maximum height h:
h(2.75) = -16(2.75)^2 + 88(2.75) + 4
h(2.75) = -16(7.5625) + 242 + 4
h(2.75) = -121 + 242 + 4
h(2.75) = 125
So, the maximum height the rocket will reach is 125 feet.
To find the time it takes for the rocket to reach its maximum height, we can use the formula for the vertex of a quadratic function, which is given by -b/2a. In the equation h(t) = -16t^2 + 88t + 4, we can see that a = -16 and b = 88.
Using the formula, the time it takes for the rocket to reach its maximum height is:
t = -b / (2a)
t = -88 / (2 * -16)
t = -88 / -32
t = 2.75 seconds
So, it will take the rocket 2.75 seconds to reach its maximum height.
To find the maximum height of the rocket, we substitute the value of t back into the equation h(t):
h(t) = -16t^2 + 88t + 4
h(2.75) = -16(2.75)^2 + 88(2.75) + 4
h(2.75) = -16(7.5625) + 242 + 4
h(2.75) = -120.8 + 242 + 4
h(2.75) = 125.2
Therefore, the maximum height of the rocket is 125.2 feet.