Question

A soccer ball is kicked toward the goal. The height of the ball is modeled by the function h(t) = −16t2 + 48t, where t equals the time in seconds and h(t) represents the height of the ball at time t seconds. What is the axis of symmetry, and how does it relate to the time the ball is in the air?

a
t = 3; It takes the ball 3 seconds to reach the maximum height and 3 more seconds to fall back to the ground.

b
t = 3; It takes the ball 3 seconds to reach the maximum height and 6 more seconds to fall back to the ground.

c
t = 1.5; It takes the ball 1.5 seconds to reach the maximum height and 3 more seconds to fall back to the ground.

d
t = 1.5; It takes the ball 1.5 seconds to reach the maximum height and 1.5 more seconds to fall back to the ground.

Answer 1

The correct answer is:

C

t = 1.5; It takes the ball 1.5 seconds to reach the maximum height and 3 more seconds to fall back to the ground.

Step-by-step explanation:

In the given function h(t) = -16t^2 + 48t, the term -16t^2 represents the downward trajectory of the ball due to gravity, and 48t represents the initial upward velocity.

To find the axis of symmetry, we use the formula t = -b / (2a), where a and b are the coefficients of the quadratic equation. In this case, a = -16 and b = 48.

Plugging these values into the formula, we get:

t = -48 / (2 * -16)

t = -48 / -32

t = 1.5

Therefore, the axis of symmetry is t = 1.5. This means that the ball reaches its maximum height after 1.5 seconds and then begins to descend. After reaching the maximum height, it takes an additional 1.5 seconds to fall back to the ground. So, in total, it takes 1.5 seconds to ascend and 1.5 seconds to descend, making a total of 3 seconds in the air.

Answer 2

The axis of symmetry of a parabola represented by the equation h(t) = -16t^2 + 48t is given by the formula t = -b/2a, where a and b are coefficients of the quadratic equation.

In this case, a = -16 and b = 48. Plugging these values into the formula, we have:

t = -b / (2a) = -48 / (2 * -16) = -48 / -32 = 1.5

Therefore, the axis of symmetry is t = 1.5.

Now let's analyze the meaning of the axis of symmetry in relation to the ball's motion. The axis of symmetry represents the time at which the ball reaches its maximum height. In this case, the ball reaches its maximum height after 1.5 seconds.

Since the ball reaches the maximum height after 1.5 seconds, it will take an equal amount of time to descend from the maximum height to the ground. Therefore, the ball will take an additional 1.5 seconds to fall back to the ground.

Hence, the correct answer is:

d) t = 1.5; It takes the ball 1.5 seconds to reach the maximum height and 1.5 more seconds to fall back to the ground.

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