Please answer this

Question

The height h (in feet) of a badminton birdie t seconds after it is hit can be modeled by the function

h = - 16 t ^ { 2 } + 32 t + 4.

a. Find the maximum height of the birdie. b. How long is the birdie in the air?

Answer 1

If you sketch this equation on time (horizontal axis) by height (vertical axis) graph, it's an upside down parabola. Let's say the time is in seconds and the height is in feet.

The y-intercept is 4, which means when the person hits the birdie at time = 0, the birdie is 4 feet off the ground.

a. What is the maximum height of the birdie?

The vertex of the parabola (at the top) is the maximum height of the birdie. To get the time at the vertex, you use the vertex formula t = -b/2a = -32/2(-16) = 1 second. Then to get the height, plug 1 into the equation h = -16(1^2) + 32(1) + 4 = 20 feet.

b. How long is the birdie in the air?

The birdie starts at 4 feet above the ground but lands on the ground, where h = 0 feet. So you want the time t for this equation h = 0 = -16(t^2) + 32t + 4.

First you can simplify this by dividing both sides by 4 to get 0 = -4(t^2) + 8t +1. And then use the quadratic formula: [ -8 + or - sqrt( (-8)^2 - 4(-4)1 ) ] / 2(-4) = 1 + or - sqrt(5)/2.

So the birdie lands at t = 1 + sqrt(5)/2 seconds = 2.12 seconds rounded. That's how long it was in the air.

BTW, the other solution 1 - sqrt(5)/2 seconds is -0.12, which is backwards in time and doesn't apply to physical world.