Question

A fish jumping out of the water can be modeled by the function ℎ(t)=−1682+431t6 where ℎ(t) represents the height of the fish after t seconds. An eagle nearby spots the fish and dives to catch it; its flight path is modeled by the function ℎ(t)=27−13t. After how many seconds does the eagle catch the fish, and at what height above the water?

A) Time: 2 seconds, Height: 56 meters
B) Time: 3 seconds, Height: 12 meters
C) Time: 1 second, Height: 14 meters
D) Time: 4 seconds, Height: 0 meters

Answer 1

To find out when the eagle catches the fish, set the height functions of the fish and eagle equal to each other and solve for time. The eagle catches the fish after 3 seconds at 12 meters high, which is option B.

Step-by-step explanation:

To determine after how many seconds the eagle catches the fish and at what height above the water, we need to set the two functions for the height of the fish, h(t) = -16t^2 + 431t/6, and the flight path of the eagle, h(t) = 27 - 13t, equal to each other and solve for t. This gives us the time when both the eagle and the fish are at the same height.

• Equating the two height functions: -16t^2 + 431t/6 = 27 - 13t.
• Simplify and solve the quadratic equation for t.
• Plug the value of t back into either of the two functions to find the height h at which the eagle catches the fish.

By solving the quadratic equation, we find that the eagle catches the fish after 3 seconds, at a height of 12 meters above the water, which corresponds to option B) Time: 3 seconds, Height: 12 meters.