Final answer:
To find the times when the ball reaches heights of 40 and 60 feet, one must set the quadratic height function to these values and solve for time. The maximum height can be determined by analyzing the vertex of the parabola, which indicates whether the ball can reach 140 feet.
Step-by-step explanation:
The student is asking how to find the times at which a ball, thrown upward, reaches specific heights given a quadratic equation that models its motion. The initial conditions include a starting height of 5 feet, and an initial upward speed of 70 feet per second. In order to find the times when the ball reaches heights of 40 feet and 60 feet, we would set the equation s(t) = -16t2 + 70t + 5 equal to these heights and solve for t. This inevitably yields a quadratic equation that can be solved using the quadratic formula or by factoring (if possible). The third part of the question asks whether the ball will ever reach a height of 140 feet, and this question can be answered by finding the ball's maximum height and seeing if it exceeds 140 feet.
For example, to find when the ball is at 40 feet, we set the equation equal to 40:
-16t2 + 70t + 5 = 40,
which simplifies to
-16t2 + 70t - 35 = 0.
After solving this equation, we obtain two values of t, which represent the times at which the ball is at 40 feet on the way up and on the way down. Similarly, the process is repeated for a height of 60 feet. To answer part (c), we would need to analyze the vertex of the parabola represented by the quadratic equation to determine the ball's maximum height.