using quadratic equation

Question

A firework is launched off a platform 5ft above the ground. The initial velocity of the rocket was 150 feet per second. A function was created to model rocket’s distance from the point it was launched to the point it hit the ground: y= -16x^2+150x+5

All x-values represent time in seconds and all y-values represent the distance traveled by the rocket in feet.

1. What is the vertex of your graph? (Round to the nearest ones place)
2. What was the maximum height of the rocket? (Round to the nearest ones place)
3. Looking at the equation of the rocket’s path, is it factor able?
4. Use your graph to estimate how long it took your rocket to hit the ground. (Round to the nearest tenths place)

Answer 1

1. The vertex of a parabola occurs when x = -b/2a. In this case, b = 150, and a = -16. Therefore:

x = -(150) / 2* (-16) =-150/-32 = 4.685

By rounding this number to the nearest ones place, you get 4.7

2. The maximum height of the rocket occurs at the vertex. We know this because a is negative, and thus the parabola is facing downwards. To get the maximum height, plug x into the equation and solve for y:

y = -16(4.685)^2+150(4.685)+5 = 356.5624

By rounding this number to the nearest ones place, you get 356.6 feet

3. The easiest way to see if a quadratic equation is factorable is to look at the discriminant, which is b^2-4ac. The equation will only be factorable if the discriminant is a square number. In this case:

(150)^2 - 4 * (-16) * (5) = 22500 + 320 = 22820

22820 is not a square number, and therefore the equation is not factorable.

4. The rocket will hit the ground at the second y-intercept. Thus, we can set y=0 and solve for x.

0 = -16x^2 + 150 x + 5

By using the quadratic equation:

x = (-b+sqrt(b^2-4ac))/2a = (-150 +sqrt( (150)^2 - 4 * (-16) * (5)) /2*-16 = 0.0332156563

By rounding to the nearest tenths place, you get 0.03 seconds.