that the swimmer needs. The time for the swimmer to travel D 1 is T 1 = (because she swims at 4 m/s), and the time for the swimmer to travelD 2 is T 2 = . Therefore, the total time is T = +300 − . Now we simply take the derivative: = − = . Next, we set this equal to zero and solve: =
0. The best way to solve this is to move the to the other side of the equals sign and cross-
multiply:Next, we square both sides: 36x^2 = 16(250,000 + x^2 ).Simplify: 20x^2 = 4,000,000.Solve for x: x = ±447.214 m. (We can ignore the negative answer.) Therefore, she should land
1,800 − 447.214 = 1,352.786 meters from the cottage. We could verify that this is a minimum
by taking the second derivative but that would be messy. It is simpler to use the calculator to
check the sign of the derivative at a point on either side of the answer, or to graph the equation
for the time.6.
First, let’s draw a picture.