baseball is hit at a height of 3 feet above the ground. At a horizontal distance of 210 feet from home plate, the ball reaches its maximum height of 60 feet.
a)write a quadratic function to represent the height of the baseball as a function of it's distance from home plate
B)determine the height of the baseball when it is 180 feet from home plate. round the answer to the nearest 10th
The vertex is on the axis of symmetry, so
h(x) = a(x-210)^2 + k
You know that
h(210) = 60, so k=60
h(0) = 3, so that lets you find a
Now use that to answer (b)
To write a quadratic function that represents the height of the baseball as a function of its distance from home plate, we can use the vertex form of a quadratic equation:
h(x) = a(x - h)^2 + k
Where:
h(x) is the height of the baseball
x is the distance from home plate
a is the coefficient that determines the shape of the parabola
(h, k) are the coordinates of the vertex
Since the ball reaches its maximum height of 60 feet at a horizontal distance of 210 feet from home plate, we can plug in these values to find the vertex. We know that the vertex has the form (h, k), so in this case, (210, 60).
Plugging the values into the vertex form equation, we get:
h(x) = a(x - 210)^2 + 60
Now, let's find the value of 'a'. We know that at a horizontal distance of 0 feet (home plate), the ball is hit at a height of 3 feet above the ground. So, plugging in these values, we get:
3 = a(0 - 210)^2 + 60
3 = a(210^2) + 60
3 = a(44100) + 60
Subtracting 60 from both sides, we have:
-57 = a(44100)
Dividing both sides by 44100, we get:
a = -57/44100
Now we can plug this value of 'a' back into our equation:
h(x) = (-57/44100)(x - 210)^2 + 60
Therefore, the quadratic function to represent the height of the baseball as a function of its distance from home plate is:
h(x) = (-57/44100)(x - 210)^2 + 60
Now, let's determine the height of the baseball when it is 180 feet from home plate:
Plugging x = 180 into our equation:
h(180) = (-57/44100)(180 - 210)^2 + 60
h(180) = (-57/44100)(-30)^2 + 60
h(180) = (-57/44100)(900) + 60
Evaluating the expression:
h(180) = (-51300/44100) + 60
h(180) = -1.16 + 60
Rounding the answer to the nearest tenth, we get:
h(180) ≈ 58.8 feet
Therefore, the height of the baseball when it is 180 feet from home plate is approximately 58.8 feet.