A model rocket is launched from a raised platform...

Question

Its height in feet is given by h(t) = -8t^2 +288t + 40 (t = seconds after launch.)

A: After how many seconds does the object reach its maximum height?

B: What is the maximum height?

Answer 1

We are given that the height in feet of a model rocket launched from a platform is given by h(t) = -8t² +288t + 40 where the t is the time in seconds after launch.

Firstly, understand that the equation h(t) = -8t² +288t + 40 is a quadratic equation (highest power is 2). We know that when we graph a quadratic equation, it looks like a parabola or a u-shaped curve that can either open upward or downward. Basically the graph would look like a U or an upside U. This would mean that it could either have a maximum point or minimum point. We know this equation has a maximum point since it is in standard form which is y = ax² + bx + c and if it is in standard form and a < 0 and in this case a is -8, then the parabola opens down so it would have a maximum. Also, whenever the function is quadratic, we can always use the vertex formula x = -b / 2a to find the vertex which is basically the peak (maximum point) of the graph.

a) To find out after how many seconds did the rocket reach its maximum height, we can find the vertex of this quadratic equation by using x = -b / 2a and since this equation is in terms of t, we can change the x to t so t = -b / 2a:

t = -b / 2a (The b is 288 and a is -8)

= -288 / 2(-8)

= -288 / -16

= 18

So, it would take the rocket 18 seconds to reach its maximum height.

b) To find its maximum height, we can plug in t = 18 into the function h(t) since we know that the function h(t) tells us the height of the rocket at any time so since we already know that the rocket is at its peak at 18 seconds, plugging t = 18 gives us the height at the highest moment:

h(t) = -8t² + 288t + 40

h(18) = -8(18)² + 288(18) + 40 (Plugged in 18 for t)

= -8(324)+ 5184 + 40 (Simplifying)

= -2592 + 5184 + 40

= 2632

So, the rocket reaches a maximum height of 2632 units (units isn't mentioned whether it's feet, meters, etc).

Answer 2

Given:

Height of the rocket as a function of time

h(t) = -8t^2 +288t + 40

Find:

a) time for it to reach max height

b) max height

Solution:

a)

The rocket reaches maximum height when the velocity is zero

To find the velocity function we take the derivative of the height function

V(t) = -16t+288

We now set the function equal to zero

0 = -16t+288

t = 18 seconds

Maximum velocity is achieved at this time

b) What is the maximum height ?

Using the time from part (a)

We plug this time into our height function

h(18) = -8 * (18)^2 + 288 * (18) + 40

h(18) = 2632 ft