Question

Given , where s(t) is in feet and t is in seconds, find each of the following.

Answer 1

1. Vertex: (3.125, 318.75)

2. X-intercepts: -2.5 and 8

3. Maximum value: 318.75 feet

4. Average rate of change: 260 ft/s over [0, 5]

5. Instantaneous rate of change at t = 3: 268 ft/s

Explanation:

The vertex of the parabola is determined by the formula
`\(-(b)/(2a)\)`, where (a), (b), and (c) are the coefficients of the quadratic equation
`\(s(t) = -16t^2 + 100t + 200\)`. In this case, the vertex is located at (t = 3.125), and by substituting this value back into the original equation, we find the corresponding height or (s(t)) value of (318.75) feet.

To identify the x-intercepts, we set (s(t)) equal to zero and solve for (t), resulting in (t = -2.5) and (t = 8). The maximum value of the function is equivalent to the y-coordinate of the vertex, which is (318.75) feet.

The average rate of change over the interval ([0, 5]) is determined by finding the difference in (s(t)) values at (t = 5) and (t = 0), and dividing by the difference in (t) values, resulting in an average rate of change of (260) feet per second. The instantaneous rate of change at (t = 3) is found by evaluating the derivative of (s(t)) at that specific time, yielding an instantaneous rate of change of (268) feet per second.

Full Question:

Given the function
`s(t) = -16t^2 + 100t + 200`, where s(t) is measured in feet and t is measured in seconds, find the following:

The vertex of the parabola

The x-intercepts

The maximum or minimum value of the function

The average rate of change of the function over the interval [0, 5]

The instantaneous rate of change of the function at time t = 3