A golf ball is struck by a 60-degree golf club at an initial velocity of
84
feet per second. The height of the golf ball in feet is given by the quadratic function h(x)= 16 x^2/(42)^2+72.7/42x where x is the horizontal distance of the golf ball from the point of impact. What is the horizontal distance of the golf ball from the point of impact when the ball is at its maximum height? What is the maximum height obtained by the golf ball?
I think you are missing a minus sign on the x^2 term, since g = -32 ft/s^2
h(x) = 1/42 (-16/42 x^2 + 72.7x)
= x/42 (-16/42 x + 72.7)
h=0 when x=0 (ball is hit), and when
x = 190.8 (ball lands)
The symmetry of a parabola means that the ball is at its highest midway between the two roots.
Just plug that in to find the max height.
http://www.wolframalpha.com/input/?i=-16+x^2%2F%2842%29^2%2B72.7%2F42x
330.2657
To find the horizontal distance of the golf ball from the point of impact when it is at its maximum height, we need to find the x-coordinate of the vertex of the quadratic function h(x).
The vertex of a quadratic function in the form h(x) = ax^2 + bx + c is given by the x-coordinate:
x = -b / (2a)
In this case, the function is h(x) = 16x^2 / 42^2 + 72.7 / 42x.
Comparing this to the standard form ax^2 + bx + c, we can see that a = 16 / 42^2, b = 72.7 / 42, and c = 0 (since there is no constant term).
Using the formula for the x-coordinate of the vertex, we can substitute these values:
x = -(72.7 / 42) / (2 * 16 / 42^2)
Simplifying the expression:
x = -(72.7 / 42) / (32 / 1764)
x = -72.7 * 1764 / (42 * 32)
x = -31330.8 / 1344
x ≈ -23.3
Therefore, the horizontal distance of the golf ball from the point of impact when it is at its maximum height is approximately 23.3 feet.
Now, let's calculate the maximum height obtained by the golf ball.
To find the maximum height, we substitute the x-coordinate of the vertex into the quadratic function and solve for h(x).
h(x) = 16x^2 / 42^2 + 72.7 / 42x
h(-23.3) = 16(-23.3)^2 / 42^2 + 72.7 / 42(-23.3)
Calculating this expression:
h(-23.3) ≈ 0.458
Therefore, the maximum height obtained by the golf ball is approximately 0.458 feet.
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.