Question

A rocket is launched from 5 feet above the ground. The height of the rocket above the ground follows the equation h(t)= -16t ² +160t +5 where t is time in seconds and h(t) is height in meters.

What is the highest point the rocket reaches during its flight? How long does it take to get to that height?
You're going to need to show work in the area below.
O The rocket reaches a height of 405 meters and it takes 10 seconds to get there
O The rocket reaches a height of 245 meters and it takes 7.5 seconds to get there
O The rocket reaches a height of 305 meters and it takes 5 seconds to get there
O The rocket reaches a height of 520 meters and it takes 6 seconds to get there"

Answer 1

The rocket reaches a height of 405 meters at its peak, and it takes 5 seconds to reach that height, as calculated using the vertex formula for the quadratic equation representing the rocket's flight.

Step-by-step explanation:

To find the highest point the rocket reaches during its flight, we need to maximize the function h(t) = -16t² + 160t + 5. This quadratic equation represents the height (in meters) of the rocket above the ground at any time t (in seconds). The highest point, or the vertex of the parabola, can be found using the formula t = -b/(2a), where a is the coefficient of t² and b is the coefficient of t.

Plugging in the values from our equation, we get: t = -160 / (2 * -16) = 5 seconds. This is the time it takes for the rocket to reach its maximum height.

Next, we can substitute t back into the height equation to find the maximum height: h(5) = -16(5)² + 160(5) + 5 = 405 meters.

Therefore, the rocket reaches a height of 405 meters, and it takes 5 seconds to get there.