This problem has 11 parts to it

Question

An oil tanker stikes a sand bar that rips a hole in the hull of the ship. Oil begins leaking out of the tanker with the spilled oil forming a circle around the tanker. The radius of the circle is increasing at the rate of 1.8 feet per hour. Round answers to two decimal places,if necessary.

1. Write the area of the circle as a function of the radius, r. Call the function A(r).

2. Write the radius of the circle as a function of time,t. Call the function r(t).

3. What is the radius of the circle after 3 hours? and what is the radius after 3.5 hours?

4.Use the result of part 3 to determine the area of the circle after 3 hrs. and after 3.5 hrs.

5. Determine a function that represents area as a function of time, t. Call the function A(t).

6.Use the result of part 5 to derermine the are of the circle after 3hrs and after 3.5 hrs.

7. Comput the average rate of change of the area of the circle from 3 hrs to 3.5 hrs.

8. Compute the average reate of change of the area of the circle from 4hrs to 4.5 hrs.

9. Based on the results obtained in parts 7 & 8, what is happening to the average rate of change of the area of the circle as time passes?

10 If the oil tanker is 100 yards from shore, when will the oil spill first reach shorelin?

11. How long will it be until 4 milesof shorline is contaminated with oil?

 

I hope you can help me with this thanks!!

Answer 1

(1) A=πr²

(2) r=1.8t

(3) r=1.8×3=5.4 ft after 3 hours, 6.3 ft after 3.5 hours

(4) A=5.4²π=91.61 sq ft after 3 hrs, 124.69 sq ft after 3.5 hrs

(5) A=πr²=π(1.8t)²=3.24πt²

(6) A=3.24π×9=91.61 sq ft and 3.24π×12.25=124.69 sq ft

(7) dA/dt is the instantaneous rate of change, dA/dt=6.28πt sq ft/hr at time t. 

Average rate of change = 6.28π(3+3.5)/2=20.41π=64.12 sq ft/hr.

(8) Average rate of change = 6.28π(4+4.5)=26.69π=83.85 sq ft/hr

(9) Rate of change is increasing (accelerating) at 6.28 sq ft/hr/hr

(10) 100yds=300ft. t=300/1.8=166⅔hrs=166hr 40min to reach the shore (6 days 22hrs 40min)

(11) 4 miles=21120ft. Assuming a straight shoreline, this is the length of a chord AB which is 100 yds from the ship centre O. We have an isosceles triangle AOB with base=21120ft, equal sides=r ft and height h=300ft. r=√(300²+10560²) (the height bisects the base). So r=10564.26ft approx, and t=r/1.8=5869.0336hrs=244days 13hrs 2min (about 8 months).