You are at the local beach for the day. At around 10:00 a.m., the lifeguards spot 3 Great White Sharks and close the beach to swimmers. There is no swimming in the ocean allowed after the lifeguards go home which is at 6:00 p.m. You aren’t sure if you should wait around or go home. You know that the Great Whites swim approximately 1.5 miles per hour. The lifeguards say that they won’t open the beach again until the sharks are likely out of the area. The beach is approximately 10 square miles of ocean
How long will the beach closure likely last? How much time, if any, will you have to enjoy the waves when it reopens? Explain how you figured this out.

1) Approximately 5 hours

2) Approximately 3.00 hours

Step-by-step explanation:

1) The given parameters are;

The time of day at the beach = 10:00 a.m.

The time the lifeguards go home = 6:00 p.m.

The speed of the Great Whites = 1.5 miles per hour

The area of the beach ≈ 10 miles² of ocean

Assuming a semicircular beach area

The radius of the beach, r = √(2×10/π) ≈ 2.52 miles

The beach diameter ≈ 5.05 miles

The time before there will be no swimming = 18:00 - 10:00 = 8 hours

Let x be the distance swam by the Great White such that they don't get back to the beach before 8 hours

Total distance swam by the Great White = x + x - 2.52 = 2·x - 2.52

Time = Distance/Speed

Which gives;

8 = (2·x - 2.52)/1.5

1.5 × 8 = 2·x - 2.52

12 = 2·x - 2.52

x = (12+2.52)/2 = 7.26

x = 7.26 miles

The time it takes the Great White to swim 7.26 miles is t = 7.26/1.5  = 4.84 hours

Therefore, the beach closure would likely last for 4.84 hours which is approximately 5 hours

2) The time remaining to enjoy the waves when it reopens is given as follows;

Time remaining = 18:00 - (10:00 + 5.00)  ≈ 3.00 hours.

(See explanation)

Step-by-step explanation:

The differential equation for a exponential growth is:

The formula is rearranged and integrated afterwards:

The exponential growth function can be used to find the montly growth function as follows:

The coeffcients of the expression are, respectively:

The exponential growth function with a monthly growth rate is:

, where t is measured in months.