the height of a rocket above the ground is modelled by the quadratic function h(t)=-4t^2+32t

a) how long will the rocket be in the air ? how do you know?

b) what is the maximum height that the rocket will reach ?

h(x) = 4t(8-t)

Since it leaves the earth at t=0 and comes back at t=8, I'd say 8 units (presumably seconds).

It reaches max height at (0+8)/2 = 4.
h(4) = 16(4) = 64

a) Well, to know how long the rocket will be in the air, we need to find the time when the height, h(t), will be zero. So let's set h(t) = 0 and solve for t.

0 = -4t^2 + 32t

Now, to solve this equation, we can either use factoring or the quadratic formula. But hey, let's make things interesting and try a joke instead!

Why did the quadratic equation go to the party without its solution?

Because it couldn't find a real root cause the party had imaginary numbers only! 🎉

Alright, let's get back to solving the equation!

Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), and in our case, a = -4, b = 32, and c = 0.

t = (-32 ± √(32^2 - 4(-4)(0))) / (2(-4))
t = (-32 ± √(1024)) / (-8)
t = (-32 ± 32) / (-8)

Using both the plus and minus signs, we get two possible values for t:
t1 = (32 - 32) / (-8) = 0
t2 = (-32 - 32) / (-8) = 8

Since we can't have negative time, we disregard t2. Therefore, the rocket will be in the air for 8 seconds. And no, I didn't use my clown magic to make time disappear. It's just simple math!

b) To find the maximum height that the rocket will reach, we need to find the vertex of the quadratic function. The x-coordinate of the vertex will give us the time when the rocket reaches its maximum height. So let's use the formula -b / (2a) to find t:

t = -32 / (2(-4))
t = -32 / (-8)
t = 4

Now, let's substitute t = 4 into the function h(t) to find the maximum height:

h(4) = -4(4)^2 + 32(4)
h(4) = -4(16) + 128
h(4) = -64 + 128
h(4) = 64

So, the maximum height that the rocket will reach is 64 units above the ground. That's pretty high! Just like the rocket's aspirations!

And remember, when it comes to math, always aim for the sky! ✨

To find the answers to your questions, we can analyze the given quadratic function.

a) To determine how long the rocket will be in the air, we need to find the time when the height (h) of the rocket is zero. This is because the rocket will land when it reaches ground level. We can set the equation equal to zero and solve for t:

-4t^2 + 32t = 0

Factoring out common terms:

t(-4t + 32) = 0

Now set each factor equal to zero:

t = 0
-4t + 32 = 0

From the first equation, we see that t = 0. This means the rocket was on the ground when time started. However, for the rocket to have any height, t must be greater than zero. Thus, we need to solve the second equation to find the positive value of t:

-4t + 32 = 0
-4t = -32
t = 8

Therefore, the rocket will be in the air for 8 seconds.

b) To find the maximum height the rocket will reach, we need to determine the vertex of the quadratic function. The vertex can be found using the formula t = -b/2a, where the quadratic function is in the form of h(t) = at^2 + bt + c.

Comparing our given function h(t) = -4t^2 + 32t to the general form, we can see that a = -4 and b = 32.

t = -b/2a
t = -32 / (2 * -4)
t = 8 / -8
t = -1

Now we can substitute t = -1 into the function to find the maximum height:

h(t) = -4 (-1)^2 + 32 (-1)
h(t) = -4 + (-32)
h(t) = -36

Therefore, the maximum height the rocket will reach is 36 units above the ground.

To find the answers to the given questions, we will use the quadratic function h(t) = -4t^2 + 32t, where h(t) represents the height of the rocket above the ground at time t.

a) How long will the rocket be in the air? How do you know?

To determine the time the rocket will be in the air, we need to find the values of t for which the height of the rocket, h(t), is greater than zero. When the height of the rocket is zero, it means the rocket has landed and is no longer in the air.

So, we set h(t) = -4t^2 + 32t > 0.
Factor out common terms: -4t(t - 8) > 0.

Now, you can use the zero-product property to find the values of t that satisfy the inequality. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero.

-4t(t - 8) = 0
Either -4t = 0 or (t - 8) = 0.

Solving each equation:

-4t = 0
t = 0

t - 8 = 0
t = 8

The values of t that satisfy the inequality -4t(t - 8) > 0 are t < 0 or t > 8.

Since time cannot be negative, the rocket is in the air for t > 8 units of time.

b) What is the maximum height that the rocket will reach?

To find the maximum height, we need to identify the vertex of the quadratic function, which represents the highest point on the parabolic curve.

The formula for the vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by:
t = -b / (2a)

For our quadratic function h(t) = -4t^2 + 32t, a = -4 and b = 32. Substituting these values into the formula:
t = -32 / (2(-4))
t = -32 / (-8)
t = 4

The maximum height occurs at time t = 4.

To find the corresponding height, substitute t = 4 into the h(t) function:
h(4) = -4(4)^2 + 32(4)
h(4) = -4(16) + 128
h(4) = -64 + 128
h(4) = 64

Therefore, the maximum height that the rocket will reach is 64 units above the ground.