Please help me figure out this quadratic equation word problem?

Question

Suppose you throw a ball straight up from the ground with a velocity of 180 ft per second. As the ball moves upward, Gravity slows it. Eventually the ball begins to fall back to the ground. The height H of the ball after t seconds in the air is given by the quadratic function h(t)= -16t^2+80t. How high does the ball go? How many second does it take for the ball to hit the ground?

Answer 1

I'm going to assume that it should be -16t^2 + 180t since you said the initial velocity was 180 but it will work either way just change 180 to 80 and solve.

 

This problem is basically asking you to find the vertex of the parabola(as this is the time it takes to reach this max height before it comes back down) remember that the vertex formula is -b/2a for any quadratic of the form ax^2 + bx + c

 

So our quadratic is -16x^2 + 180x so our a = -16, our b is 180, and our c is 0. Plugging this into -b/2a gives:

 

-180/-32 which equals 5.625 seconds. After solving for this time, plug it back into the h(t) so you get:

 

h(5.625) = -16*(5.625)^2 + 180*(5.625) = 506.25 feet above the ground.

 

For the second part, you can abuse the symmetry of a parabola to find the time. Since it took 5.625 seconds to reach the top, and it's the exact same distance from the top to the bottom. It will take exactly 2*(5.625) to get back to the ground.

Answer 2

Set H = 0 and solve for t. Here's why: Your ball is at height = 0 TWICE, at the start and at the end of its journey through the air (when you launch and when it lands.) Since the time up = time down, you can find the second solution for t (the first solution for t is zero, the time of launch) to find out how long it took to land... AND if you divide that total time by 2, you will know how much time it took for the ball to reach the top of its arc (when velocity became zero.)

 

Now solve the problem again. Take that total time divided by two and substitute it for t in the equation. You are going to solve the equation to find H, the maximum height of the ball.

 

Good luck!