A ball is hit from the ground. When the ball has traveled a horizontal distance of d meters, its height, h, in meters, can be modeled by the function h(d) = 1/125d^2 + d
What is the horizontal distance from the point where the ball is hit to the point where the ball lands on the ground?
When the ball lands on the ground, its height will be 0. We can set h(d) = 0 and solve for d:
0 = 1/125d^2 + d
Multiplying both sides by 125d^2, we get:
0 = d^2 + 125d
Now we can use the zero product property and set each factor equal to zero:
d^2 = 0 or d + 125 = 0
The first solution d = 0 doesn't make sense in this context because it would mean the ball didn't travel any distance horizontally. Therefore, we use the second solution:
d + 125 = 0
d = -125
This negative value doesn't make sense either, so we conclude that the ball lands on the ground at a horizontal distance of 125 meters from where it was hit.
To find the horizontal distance from the point where the ball is hit to the point where the ball lands on the ground, we need to determine the value of d when the height of the ball, h(d), is equal to 0.
Given the function h(d) = 1/125d^2 + d, we can set it equal to 0:
0 = 1/125d^2 + d
To solve this equation, we can multiply both sides by 125 to eliminate the fraction:
0 = d^2 + 125d
Now, we can factor out d:
0 = d(d + 125)
Setting each factor equal to 0 gives us two possible solutions:
d = 0
d + 125 = 0
The first solution, d = 0, represents the initial point where the ball is hit from the ground. Since we're interested in the distance the ball travels, this is not the answer we're looking for.
The second solution, d + 125 = 0, gives us:
d = -125
The negative value does not make physical sense in this context, as we're dealing with distances. Therefore, we can discard this solution as well.
This means that there is no real value of d where the height of the ball is equal to 0, which implies that the ball does not land on the ground.