Every year in Delaware there is a contest where people create cannons and catapults designed to launch pumpkins as far in the air as possible. The equation y = 12 + 105x - 16x ^ 2 can be used to represent the height, y, of a launched pumpkin, where x is the time in seconds that the pumpkin has been in the air. What is the maximum height that the pumpkin reaches? How many seconds have passed when the pumpkin hits the ground? (Hint: If the pumpkin hits the ground, its height is 0 feet) (1 point)
To find the maximum height that the pumpkin reaches, we need to find the vertex of the quadratic equation y = 12 + 105x - 16x^2.
First, let's rewrite the equation in vertex form:
y = -16x^2 + 105x + 12.
The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) is the vertex.
Comparing the given equation to the vertex form, we can see that a = -16.
To find the x-coordinate of the vertex, we can use the formula h = -b / 2a.
In this case, b = 105 and a = -16:
h = -105 / (2 * -16) = -105 / -32.5 = 3.23 (approximately).
So the pumpkin reaches its maximum height at x = 3.23 seconds.
To find the y-coordinate (maximum height) of the vertex, we substitute this x-value back into the original equation:
y = -16(3.23)^2 + 105(3.23) + 12.
Calculating this expression, we find:
y ≈ 196.15.
Therefore, the maximum height the pumpkin reaches is approximately 196.15 feet.
To determine how many seconds have passed when the pumpkin hits the ground (height is 0 feet), we set y = 0 and solve the quadratic equation:
0 = -16x^2 + 105x + 12.
We can either factor this equation or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
In this case, a = -16, b = 105, and c = 12:
x = (-105 ± √(105^2 - 4(-16)(12))) / (2 * -16).
Calculating this expression, we find two solutions:
x ≈ 0.064 and x ≈ 8.19.
Since time cannot be negative in this context, we discard the solution x ≈ 0.064.
Thus, the pumpkin hits the ground at approximately 8.19 seconds.
In summary:
- The maximum height that the pumpkin reaches is approximately 196.15 feet.
- The pumpkin hits the ground after approximately 8.19 seconds.