A toy rocket is shot vertically into the air from a launching pad 6 feet above the ground with an initial velocity of 128 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)=−16t2+128t+6. 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 find the vertex of the quadratic function h(t) = -16t^2 + 128t + 6. The t-coordinate of the vertex can be found using the formula t = -b/(2a), where a, b, and c are coefficients of the quadratic function in the form ax^2 + bx + c. In this case, a = -16 and b = 128.
t = -b/(2a) = -128/(2*(-16)) = 128/32 = 4
So, it will take 4 seconds for the rocket to reach its maximum height.
To find the maximum height, we substitute this value of t into the quadratic function:
h(4) = -16(4)^2 + 128(4) + 6 = -256 + 512 + 6 = 262
Therefore, the maximum height of the rocket is 262 feet.
To find the time it takes for the rocket to reach its maximum height, we need to determine when the velocity of the rocket becomes zero.
We know that the velocity of an object at any time t is given by the derivative of the height function. Let's find the derivative of h(t):
h(t) = -16t^2 + 128t + 6
Taking the derivative with respect to t, we get:
h'(t) = -32t + 128
Now, we set h'(t) equal to zero and solve for t:
-32t + 128 = 0
-32t = -128
t = -128 / -32
t = 4
Therefore, the rocket will reach its maximum height at t = 4 seconds.
To find the maximum height, we substitute this value of t back into the height function h(t):
h(4) = -16(4)^2 + 128(4) + 6
h(4) = -16(16) + 512 + 6
h(4) = -256 + 512 + 6
h(4) = 262
The maximum height of the rocket is 262 feet.
To find the time it takes for the rocket to reach its maximum height, we can use calculus. The maximum height occurs at the vertex of the parabolic function h(t) = -16t^2 + 128t + 6.
The t-coordinate of the vertex of a parabola can be found using the formula -b/2a, where a and b are the coefficients of the quadratic equation in standard form (at^2 + bt + c = 0).
In this case, a = -16 and b = 128. So, the formula becomes -128/(2*(-16)) = -128/(-32) = 4 seconds.
Therefore, it takes the rocket 4 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t = 4 into the equation for h(t):
h(4) = -16(4)^2 + 128(4) + 6
= -16(16) + 512 + 6
= -256 + 512 + 6
= 262 feet
So, the maximum height reached by the rocket is 262 feet.