To find the horizontal distance of the golf ball from the point of impact when it reaches its maximum height, we need to identify the x-value at which the height function h(x) reaches its maximum.
The height function is given by h(x) = (16x^2)/(42)^2 + (72.7/42)x.
To find the maximum height, we can rearrange the equation of the function h(x) in vertex form. The vertex form of a quadratic function is given by h(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Expanding the function h(x) into vertex form, we have:
h(x) = (16/42^2)(x^2) + (72.7/42)x
= (16/1764)x^2 + (72.7/42)x
We can rewrite the function as:
h(x) = (16/1764)(x^2) + (72.7/42)x + 0
Comparing this form to the vertex form h(x) = a(x - h)^2 + k, we can determine the vertex coordinates: h = -b/2a and k = f(h), where f(h) is the value of h(x) at the x-coordinate h.
In this case, a = 16/1764, b = 72.7/42, and c = 0. Therefore, we have:
h = -b/2a = -((72.7/42)/(2*(16/1764)))
= -((72.7/42)/(32/1764))
= -((72.7/42)*(1764/32))
= -((72.7*1764)/(42*32))
= -46.875
So, the x-coordinate of the vertex is -46.875, indicating that the maximum height is reached when the ball is approximately 46.875 feet from the point of impact.
To find the maximum height obtained by the golf ball, we substitute the x-coordinate of the vertex (-46.875) back into the function h(x).
Therefore, the maximum height is given by h(-46.875) = (16/1764)(-46.875)^2 + (72.7/42)(-46.875)
Solving this expression will give us the maximum height obtained by the golf ball.