Question

50 POINTS !!!!

Fernando launches a toy rocket into the air. The rocket's height, y, in meters with respect to time, x, in seconds, can be modeled by the function
`y=-4x^2+4x+24`. What is the maximum height, in meters, of the rocket?

Must provide a thorough step-by-step explanation.
I have the answer key and the right answer I just need the step-by-step.

Answer 1

Explanation:

Hello there, To find the maximum height of Fernando's toy rocket, we'll use the equation y = -4x^2 + 4x + 24. The vertex of a quadratic function is the highest or lowest point on its graph, and that's exactly what we need to find.

First, let's identify the coefficients of the quadratic equation. In this case, the coefficient of x^2 is -4, the coefficient of x is 4, and the constant term is 24.

Now, we'll use the formula x = -b/2a to find the x-coordinate of the vertex. Here, a is the coefficient of x^2 (-4) and b is the coefficient of x (4).

Plugging in the values, we get:

x = -4 / (2 * -4)

x = -4 / -8

x = 0.5

Now, we'll substitute the x-coordinate back into the equation to find the y-coordinate of the vertex.

Plugging in the value, we get:

y = -4(0.5)^2 + 4(0.5) + 24

y = -4(0.25) + 2 + 24

y = -1 + 2 + 24

y = 25

So, the vertex of the function is (0.5, 25). The x-coordinate represents the time (in seconds) when the rocket reaches its maximum height, and the y-coordinate represents the maximum height (in meters) of the rocket.

Therefore, the maximum height of the rocket is 25 meters.

I hope this helps! Let me know if you have any other questions.

Answer 2

25 meters

Explanation:

In order to find the maximum height of the rocket, we need to determine the vertex of the parabolic function. The vertex of a parabola in the form y = ax² + bx + c can be found using the formula:

`\sf x = (-b)/(2a)`

Once we find the value of x, we can substitute it back into the original equation to find the corresponding y, which will be the maximum height.

In this case, the function of modeling the rocket's height is y = -4x² + 4x + 24, where:

• a = -4
• b = 4
• c = 24

Let's find the x-coordinate of the vertex.

Use the formula to find the x-coordinate of the vertex:

`\sf x = \frac{\cancel{-4}}{2 \cdot\cancel{ (-4)}} = (1)/(2)`

Now,

Let's find the corresponding y-coordinate.

Now that we have
`\sf x = (1)/(2)`, substitute this value into the original equation to find the y-coordinate:

`\sf y = -4\left((1)/(2)\right)^2 + 4\left((1)/(2)\right) + 24`

`\sf y = -4\cdot(1)/(4) + 2 + 24`

`\sf y = -1 + 2 + 24`

`\sf y = 25`

So, the maximum height of the rocket is 25 meters.