Question

A person standing close to the edge on top of a 64-foot building throws a ball vertically upward. The quadratic function h = â 16t" + 48t + 64 models the ball's height about the ground, h, in feet, t seconds after it was thrown.

a) What is the maximum height of the ball?
______ feet
b) How many seconds does it take until the ball hits the ground?
______seconds

Answer 1

The maximum height is found by determining the vertex of the quadratic function, which occurs at
`\(t = (-b)/(2a) = 3\),` yielding h = 80 feet. The time until the ball hits the ground is obtained by solving h = 0, resulting in
`\(t \approx 4\)` seconds, as the negative root is disregarded.

Step-by-step explanation:

In the given quadratic function h = -16t^2 + 48t + 64, the coefficient of the squared term -16 indicates that the parabolic graph opens downward, suggesting a maximum value. To find the maximum height, we can use the vertex form of a quadratic function, h = a(t - h)^2 + k, where (h, k) is the vertex. In this case,
`\(h = (-b)/(2a)\)` gives us the time when the ball reaches its highest point. Substituting the values from the given function, we get
`\(t = (-48)/(2(-16)) = 3\)`. Plugging this back into the original function, we find the maximum height:
`\(h = -16(3)^2 + 48(3) + 64 = 80\) feet.`

To determine the time it takes for the ball to hit the ground, we set h equal to zero and solve for
`\(t\): \(0 = -16t^2 + 48t + 64\)`. Factoring or using the quadratic formula, we find two solutions. However, since time cannot be negative in this context, we discard the negative root. The positive root gives us
`\(t \approx 4\)` seconds. Therefore, it takes approximately 4 seconds for the ball to hit the ground after being thrown.

Answer 2

The maximum height is obtained 1.5 seconds after the ball was thrown, and the ball hits the ground after 4 seconds.

To find the maximum height of the ball, we use the vertex formula with the coefficients of the quadratic function. Similarly, to find the time it takes for the ball to hit the ground, we set the height to zero and solve for t.

So, to find the maximum height of the ball, we need to determine the vertex of the quadratic function. The vertex can be found using the formula :

t = -b / (2a), where a is the coefficient of the t^2 term and b is the coefficient of the t term.

In this case, a = -16 and b = 48.

Plugging in these values, we can calculate t = -48 / (2*(-16)) = 1.5.

So, the maximum height of the ball is obtained 1.5 seconds after it was thrown.

To find how long it takes for the ball to hit the ground, we need to set h = 0 and solve for t.

Substituting h = 0 into the quadratic function, we get -16t^2 + 48t + 64 = 0.

Using either factoring or the quadratic formula, we find that the roots are t = -1 and t = 4.

Since we are only interested in the positive root, the ball hits the ground after 4 seconds.