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

Answer:

Maximum height achieved by football = 25.5 yards.

Distance traveled by football  51 yards.

Step-by-step 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 $\frac{dy}{dx}$ ( derivative ) of the equation and equate it to zero.

$\frac{d(y)}{dx}$ = $\frac{d}{dx}$ (-0.03x² + 1.53x )

$\frac{dy}{dx}$ = -0.03 × 2x + 1.53

By putting $\frac{dy}{dx}$ = 0

-0.06x + 1.53 = 0

0.06x = 1.53

x = $\frac{1.53}{0.06}$ = 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 = $\frac{1.53}{0.03}$ = 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.
$x=\frac{-1.53}{2*-.03}=25.5$

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.