Question

A baseball is hit so that its height in feet, t seconds after it is hit, can be modeled by the function h(t) = -8+2 +377 - 3. A hawk takes off from its nest 28 feet above the ground at the same time that the player hits the baseball. The hawk's flight can be modeled by the function h(t) = 28 - t. After how many seconds do the hawk and baseball first achieve the same height? What is that height?

Answer 1

The baseball and the hawk achieve the same height after approximately 0.54 seconds, and the height at which they meet is about 28 feet.

Step-by-step explanation:

To find out after how many seconds the hawk and the baseball achieve the same height, we need to set the height functions equal to each other and solve for t. This gives us the equation:

hbaseball(t) = hhawk(t), which becomes -8t2 + 377 - 3 = 28 - t.

When solving this quadratic equation for t, we find two possible times: t = 3.79 s and t = 0.54 s. However, we are interested in the first time the two heights are equal, which is t = 0.54 s. At this time, the baseball is on its way up, and so is the hawk.

To find the height, we substitute t into one of the original height functions:

hbaseball(0.54) = -8(0.54)2 + 377(0.54) - 3 ≈ 28 feet.

So, the baseball and the hawk are at the same height of approximately 28 feet after about 0.54 seconds.