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?

Question

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?

GPT 3.5 · Answer

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.