Question

Emma is staying in a cottage along a beautiful and straight shoreline. A point Q on the shoreline is located 3 kilometers east of the cottage, and an island is located 2 kilometers north of Q. Emma plans to travel from the cottage to the island by some combination of walking and swimming. She can start to swim at any point P between the cottage and the point Q. If she walks at a rate of 4 km/hr and swims at a rate of 3 km/hr, what is the minimum possible time it will take Emma to reach the island?

Enter an exact answer or round to the nearest hundredth of an hour.

Answer 1

the minimum possible time for Emma to reach the island is approximately 2.9583 hours, when she starts swimming at a point 5/6 km east of the cottage.

Explanation:

We can use the concept of minimum distance to find the minimum possible time for Emma to reach the island. Let's assume that Emma walks to some point P along the shoreline and then swims to the island. We want to find the point P that minimizes the total distance that Emma has to cover.

Let the distance from the cottage to point P be x km. Then the distance from point P to the island is (3 - x) km. Using the Pythagorean theorem, we can find the distance from the cottage to the island as follows:

distance^2 = x^2 + (3 - x)^2 + 2^2

distance^2 = 10x - 6x^2 + 13

To minimize the distance, we can take the derivative of the distance equation with respect to x and set it equal to zero:

d(distance^2)/dx = 10 - 12x = 0

x = 5/6 km

Substituting x = 5/6 km into the distance equation, we get:

distance^2 = 25/3 km^2

distance = 5/sqrt(3) km

Therefore, the minimum possible time for Emma to reach the island is the time it takes her to walk to point P (distance x) plus the time it takes her to swim to the island (distance 3 - x), divided by her swimming speed:

time = x/4 + (3 - x)/3

time = (3x + 12 - 4x)/12

time = (12 - x)/4

time = (12 - 5/6)/4

time = 2.9583 hours (rounded to 4 decimal places)

Therefore, the minimum possible time for Emma to reach the island is approximately 2.9583 hours, when she starts swimming at a point 5/6 km east of the cottage.

Answer 2

Let's call the point where Emma starts swimming P. We can draw a diagram to visualize the situation:

Cottage ---------- Q ---------------- Island

x km 3 km 2 km

The distance from the cottage to Q is x km, and the distance from Q to the island is 2 km. The total distance Emma will have to travel can be expressed as:

d = √(x^2 + 2^2)

The time it will take Emma to walk the distance from the cottage to point P is:

t1 = x / 4

The time it will take Emma to swim from point P to the island is:

t2 = √((d - x)^2 + 2^2) / 3

The total time it will take Emma to reach the island can be expressed as:

T = t1 + t2

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

dT/dx = (1/4) - (d-x)/3√((d-x)^2+4) = 0

Solving for x, we get:

x = (3/5)d

Substituting this value of x back into the expression for T, we get:

T = (3/20)d + (2/5)√(5)

Rounding to the nearest hundredth, we get:

T ≈ 0.96 hours

Therefore, the minimum possible time it will take Emma to reach the island is approximately 0.96 hours.