A model rocket is launched from a roof into a large field. The path of the rocket can be modeled by the equation y = –0.04x2+ 8.3x + 4.3, where x is the horizontal distance, in meters, from the starting point on the roof and y is the height, in meters, of the rocket above the ground. How far horizontally from its starting point will the rocket land?

Question

A model rocket is launched from a roof into a large field. The path of the rocket can be modeled by the equation y = –0.04x2+ 8.3x + 4.3, where x is the horizontal distance, in meters, from the starting point on the roof and y is the height, in meters, of the rocket above the ground. How far horizontally from its starting point will the rocket land?

A. 208.02 m
B. 416.03 m
C. 0.52 m
D. 208.19 m

Answer 1

To find where the rocket will land, we need to find when y=0 (the rocket will be at ground level). So we substitute y=0 into the equation and solve for x:

0 = -0.04x^2 + 8.3x + 4.3

Multiplying by -25 to simplify:

x^2 - 207.5x - 86 = 0

Using the quadratic formula:

x = (207.5 ± sqrt(207.5^2 - 4(1)(-86))) / 2(1)

x = (207.5 ± sqrt(42906.25)) / 2

The two solutions are x = 208.02 and x = -0.52. We can discard the negative solution because we are looking for distance, not direction. Therefore, the rocket will land 208.02 meters horizontally from its starting point, so the answer is A.

Answer 2

are there any exponents in the model rocket's equation?

Answer 3

Yes, the equation for the path of the rocket involves an exponent: y = -0.04x^2 + 8.3x + 4.3. The term -0.04x^2 represents a quadratic function, where x is the independent variable and x^2 is the exponent.

Answer 4

To find the horizontal distance from the starting point where the rocket lands, we need to find the x-coordinate where y = 0.

Given the equation: y = –0.04x^2 + 8.3x + 4.3

Setting y = 0:
0 = –0.04x^2 + 8.3x + 4.3

Now, we can solve this quadratic equation to find the solutions for x using either factoring, completing the square, or the quadratic formula.

In this case, let's use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our equation: a = -0.04, b = 8.3, c = 4.3

Plugging in these values into the quadratic formula:

x = (-(8.3) ± sqrt((8.3)^2 - 4(-0.04)(4.3))) / (2(-0.04))

Simplifying:

x = (-8.3 ± sqrt(68.89 + 0.688)) / (-0.08)

x = (-8.3 ± sqrt(69.578)) / (-0.08)

Taking the positive value for x:

x = (-8.3 + sqrt(69.578)) / (-0.08)

x ≈ -0.52 meters or x ≈ 208.19 meters

Since we are looking for the distance from the starting point and the distance cannot be negative, the rocket will land approximately 208.19 meters horizontally from its starting point.

Therefore, the correct answer is D. 208.19 m.