With the given model: h=-0.045d^2+2d
where h is the height of the ball from the ground(in yards) and d is the horizontal distance of the ball(in yards) from its starting point.
A.)Pat is standing on the 17-yard line when the football passes over his head. How high up is the football at this point?
B.)The football is 8 yards above the ground when it passes over Billy. At what yard line is Billy standing when the football passes over his head?
for (a), we substitute 17 to d in the equation for the height,h:
h = -0.045d^2 + 2d
h = -0.045*(17^2) + 2(17)
h = 20.995 yards
for (b), we are given the height and we need to find the horizontal distance. substituting,
h = -0.045d^2 + 2d
8 = -0.045d^2 + 2d
0 = -0.045d^2 + 2d - 8
note that this is a quadratic equation and we can either factor it or use quadratic formula to solve for d,, here, let's just use quadratic formula:
d = [-b +- sqrt(b^2 - 4ac)]/(2a)
d = [-2 +- sqrt(2^2 - 4(-0.045)(-8))]/(2(-0.045))
d = [-2 +- sqrt(2.56)]/(-0.09)
d = [-2 +- 1.6]/(-0.09)
d = 40 yards
d = 4.44 yards
note that there are two answers. for the 4.44 yards, the ball is going up, while for the 40 yards, the ball is already going down.
hope this helps~ :)
To find the height of the football at a certain distance and the distance when the football is at a certain height, we can substitute the given values into the equation h = -0.045d^2 + 2d.
A.) To determine the height of the football when it passes over Pat's head at the 17-yard line, we substitute d = 17 into the equation:
h = -0.045(17)^2 + 2(17)
Simplifying this expression, we get:
h = -0.045(289) + 34
h = -13.005 + 34
h = 20.995
Therefore, the football is approximately 21 yards above the ground when it passes over Pat's head.
B.) To determine the yard line where Billy is standing when the football passes over his head at a height of 8 yards above the ground, we need to rearrange the equation and solve for d.
Starting with the given height h = -0.045d^2 + 2d, we set h = 8 and solve for d:
8 = -0.045d^2 + 2d
Rearranging this equation, we get:
0.045d^2 - 2d + 8 = 0
This is a quadratic equation. To solve it, we can use the quadratic formula:
d = (-b ± √(b^2 - 4ac)) / 2a
Applying this formula to our equation, where a = 0.045, b = -2, and c = 8, we get:
d = (-(-2) ± √((-2)^2 - 4(0.045)(8))) / 2(0.045)
Simplifying further:
d = (2 ± √(4 - 1.44)) / 0.09
d = (2 ± √(2.56)) / 0.09
d = (2 ± √(2.56)) / 0.09
Calculating the square root:
d = (2 ± 1.6) / 0.09
Therefore, we have two possible solutions for d:
d1 = (2 + 1.6) / 0.09
d1 = 26.6667
d2 = (2 - 1.6) / 0.09
d2 = 4.4444
So, Billy is standing approximately between the 4 and 5-yard line when the football passes over his head, based on the given information.