Question

The path of a football kicked by a field goal kicker can be modeled by the equation y = –0.03x2 + 1.53x, where x is the horizontal distance in yards and y is the corresponding height in yards. What is the football’s maximum height? Round to the nearest tenth. yds. How far is the football kicked? yds.

Answer 1

Maximum height achieved by football = 25.5 yards.

Distance traveled by football 51 yards.

Explanation:

Path of a football kicked by a field goal kicker can be modeled by the equation y = -0.03x² + 1.53x.

Where x represents horizontal distance and y represents height of the football.

For maximum, height we will find the
`(dy)/(dx)` ( derivative ) of the equation and equate it to zero.

`(d(y))/(dx)` =
`(d)/(dx)` (-0.03x² + 1.53x )

`(dy)/(dx)` = -0.03 × 2x + 1.53

By putting
`(dy)/(dx)` = 0

-0.06x + 1.53 = 0

0.06x = 1.53

x =
`(1.53)/(0.06)` = 25.5 yards

Therefore, maximum height achieved by the ball is 25.5 yards.

Now for horizontal distance covered by ball y should be 0

-0.03x² + 1.53x = 0

x ( 1.53 - 0.03x ) = 0

0.03x = 1.53

x =
`(1.53)/(0.03)` = 51 yards.

Football was kicked 51 yards.

Answer 2

This is the equation of an upside down parabola, so the maximum height will be at the vertex. To find the vertex, we'll use the formula for the x-coordinate and then solve for the y-coordinate.

`x=(-1.53)/(2*-.03)=25.5`

Plugging this into the given formula:

`y=-0.03(25.5)^2+1.53(25.5)=19.5075`
Since it asks us to round, the maximum height would be 20 yards. To find how far the football travels in total, we need to set the equation equal to 0 and find the roots:

`0=-0.03x^2+1.53x=x(-0.03x+1.53)`

Then by the zero product property (or the no zero divisors rule), we have:

`x=0` (this is the x value where the ball is kicked from)

`-0.03x+1.53=0`

`-0.03x=-1.53`

`x=51`

So the ball travels 51 yards in total.