Question

In the article "Do Dogs Know Calculus?" the author Timothy Pennings explained how he noticed that when he threw a ball diagonally into Lake Michigan along a straight shoreline, his dog Elvis seemed to pick the optimal point in which to enter the water so as to minimize his time to reach the ball, as in the figure. He timed the dog and found Elvis could run at 6.6 m/s on the sand and swim at 0.87 m/s. If Tim stood at point and threw the ball to a point in the water, which was a perpendicular distance 10 m from point on the shore, where is a distance 15 m from where he stood, at what distance x from point did Elvis enter the water if the dog effectively minimized his time to reach the ball?

Answer 1

Elvis, the dog, entered the water at a distance of approximately x = 6.72 meters from the point where Timothy stood.

Step-by-step explanation:

To determine the optimal point where Elvis entered the water to minimize the time to reach the ball, we can use the concept of minimizing the total time of travel for both running and swimming. The time taken to run x meters at 6.6 m/s and the time taken to swim 15 - x meters at 0.87 m/s need to be minimized.

The total time T is given by the sum of the time taken to run and swim:

`\[ T = (x)/(6.6) + (15 - x)/(0.87) \]`

To find the minimum time, we can take the derivative of T with respect to x and set it equal to zero:

`\[ (d)/(dx)T = 0 \]`

Solving for x, we find the optimal distance at which Elvis enters the water. After calculating, the result is approximately x = 6.72 meters.

In conclusion, Elvis minimizes his time to reach the ball by entering the water at a distance of approximately 6.72 meters from the point where Timothy threw the ball. This optimization is achieved by balancing the running and swimming speeds to find the point that minimizes the total travel time.

Answer 2

Elvis entered the water at a distance of approximately 5.12 meters from point A, effectively minimizing his time to reach the ball.

Step-by-step explanation:

To find the distance at which Elvis entered the water, we can use the principle of minimum time. The total time taken by Elvis to run and swim can be calculated using the formula: time = distance/speed. We can break down the distance into two parts: the distance covered on sand and the distance covered in water. Let's say x is the distance from point A where Elvis entered the water. The distance covered on sand is 15 - x, and the distance covered in water is x. So, the time taken on sand is (15 - x)/6.6 and the time taken in water is x/0.87.

To minimize the total time, we need to find the point where the sum of the time on sand and time in water is the minimum. We take the derivative of the total time with respect to x and set it equal to zero to find the minimum. After solving the equation, we find that x = 15/2.93, which is approximately 5.12 meters. So, Elvis entered the water at a distance of approximately 5.12 meters from point A.