Question

Mikala initially launches a toy rocket from a height of 3 feet above the ground. After 2 seconds, the rocket reaches its highest point of 11 ft and then returns to
the ground
.
• Create a model that can be used to determine the height of the toy rocket at a given time. Define your variables and show your work or explain how your
equation models the situation.
• Use the model to determine at what time the toy rocket will land on the ground. Show your work or explain your answer.
Enter your answers and explanations in the box.

Answer 1

• Given the vertex, the model is given by:

$y = -2x^2 + 8x + 3$

• Using the model above, it is found that it lands on the ground after 4.35 seconds.

The equation of a quadratic function of vertex (h,k) is given by:

$y = a(x - h)^2 + k$

The vertex is the maximum point, which is (2,11), hence $h = 2, k = 11$. Then:

$y = a(x - 2)^2 + 11$

The initial height is of 3 feet, then when $x = 0, y = 3$, and this is used to find a.

$y = a(x - 2)^2 + 11$

$3 = 4a + 11$

$4a = -8$

$a = -\frac{8}{4}$

$a = -2$

Then:

$y = -2(x - 2)^2 + 11$

In standard format, the model is:

$y = -2x^2 + 8x - 8 + 11$

$y = -2x^2 + 8x + 3$

It hits the ground when $y = 0$, so:

$-2x^2 + 8x + 3 = 0$

$2x^2 - 8x - 3 = 0$

Which has coefficients $a = 2, b = -8, c = -3$. So

$\Delta = b^2 - 4ac = (-8)^2 - 4(2)(-3) = 88$

$x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{8 + sqrt{88}}{4} = 4.35$

$x_1 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{8 - sqrt{88}}{4} = -0.35$

Time is positive, so it lands on the ground after 4.35 seconds.

A similar problem is given at https://brainly.com/question/17987697