Question

a ball is thrown directly upward from a height of 7 ft with an initial velocity of 24 ft/sec. the function ( s(t) = -16t^2 + 24t + 7 ) gives the height of the ball, in feet, ( t ) seconds after it has been thrown. determine the time at which the ball reaches its maximum height and find the maximum height. the ball reaches its maximum height of (square) ft (square) sec(s) after the ball is thrown. (type integers or decimals.)

Explanation:

Step1: Identify the vertex of the parabola

The height function \( s(t) = -16t^2 + 24t + 7 \) is a quadratic function in the form \( f(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 24 \), and \( c = 7 \). The time at which the maximum height occurs is at the vertex of the parabola. The formula for the \( t \)-coordinate of the vertex of a quadratic function \( f(t) = at^2 + bt + c \) is \( t = -\frac{b}{2a} \).

Substituting \( a = -16 \) and \( b = 24 \) into the formula:
\( t = -\frac{24}{2 \times (-16)} \)
\( t = -\frac{24}{-32} \)
\( t = \frac{24}{32} \)
\( t = 0.75 \) seconds.

Step2: Find the maximum height

Now that we have the time \( t = 0.75 \) seconds, we substitute this value back into the height function \( s(t) \) to find the maximum height.

\( s(0.75) = -16(0.75)^2 + 24(0.75) + 7 \)

First, calculate \( (0.75)^2 = 0.5625 \)

Then, \( -16 \times 0.5625 = -9 \)

Next, \( 24 \times 0.75 = 18 \)

Now, substitute these values back into the equation:

\( s(0.75) = -9 + 18 + 7 \)

\( s(0.75) = 9 + 7 \)

\( s(0.75) = 16 \) feet.

Answer:

The ball reaches its maximum height of \( \boldsymbol{16} \) ft \( \boldsymbol{0.75} \) sec(s) after the ball is thrown.