A ball is thrown directly upward from a height of 7 ft with an initial velocity of $$ 24 \mathrm{ft} / \mathrm{sec} $$. The function $$ \mathrm{s}(\mathrm{t})=-16 \mathrm{t}^{2}+24 \mathrm{t}+7 $$ gives the height of the ball, in feet, t seconds after it has been thrown. The ball reaches its maximum height of (Simplify your answers. Type integers or decimals.)

Question

UpStudy AI · Accepted Answer

The maximum height of the ball is 16 feet. To find the maximum height of the ball, we need to determine the vertex of the parabola represented by the function $$ s(t) = -16t^2 + 24t + 7 $$. The vertex form of a quadratic equation is $$ s(t) = a(t - h)^2 + k $$, where $$ (h, k) $$ is the vertex of the parabola. The given function is in the standard form $$ s(t) = at^2 + bt + c $$, where $$ a = -16 $$, $$ b = 24 $$, and $$ c = 7 $$. To find the vertex, we use the formula $$ h = -\frac{b}{2a} $$. Let's calculate $$ h $$: $$ h = -\frac{b}{2a} = -\frac{24}{2(-16)} = -\frac{24}{-32} = \frac{3}{4} $$ Now that we have $$ h $$, we can find the maximum height $$ k $$ by plugging $$ h $$ back into the original function: $$ k = s(h) = -16\left(\frac{3}{4}\right)^2 + 24\left(\frac{3}{4}\right) + 7 $$ $$ k = -16\left(\frac{9}{16}\right) + 24\left(\frac{3}{4}\right) + 7 $$ $$ k = -9 + 18 + 7 $$ $$ k = 16 $$ Therefore, the maximum height of the ball is $$ 16 $$ feet.

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

Related Questions