Question

Oil is leaking from an oil tanker at the rate of 10000 liters per hour. 8 liters of oil spread out over 10 square meters of ocean surface. A circular oil slick forms. (a) Express the radius R of the oil slick as a function of the time t (in hours) the tank has been leaking. If your answer involves pi type pi not 3.14 R(t)- (b) After how many hours will the oil slick have radius 1 kilometer? meters hours

Answer 1

The radius of the oil slick as a function of time is R(t) = √(8000t/π), and after approximately 392.7 hours, the oil slick will have a radius of 1 kilometer.

Step-by-step explanation:

Calculating the Radius of an Oil Slick

Given that oil spreads at a rate such that 8 liters cover 10 square meters, we have a rate of 0.8 liters per square meter. Since the oil tanker is leaking at 10000 liters per hour, the area covered by the oil slick per hour is:

Area per hour = Rate of leakage × Coverage rate = 10000 liters/hour × 0.8 m²/liter = 8000 m²/hour.

To find the radius as a function of time, use the formula for the area of a circle, A = πr², where A is the area and r is the radius.

Given A(t) = 8000t (t in hours), we solve for r(t):

πr(t)² = 8000t

Therefore, R(t) = √(8000t/π).

To find when the radius of the oil slick will be 1 kilometer, set R(t) = 1000 meters and solve for t:

1000 = √(8000t/π)

t = π(1000²)/8000

t ≈ 392.7 hours.

Answer 2

The radius of the oil slick can be expressed as a function of time using a proportion that relates the area of the oil slick to the radius. The radius of the oil slick in meters is equal to the square root of 5 divided by 200. The oil slick will have a radius of 1 kilometer after 500 hours of leaking.

Step-by-step explanation:

To express the radius of the oil slick as a function of time, we need to find the relationship between the volume of oil leaked and the radius of the slick. We know that the rate of oil leaking is 10000 liters per hour, so the volume of oil leaked after t hours is given by V(t) = 10000t. We also know that 8 liters of oil spread out over 10 square meters of ocean surface. Since a circular oil slick forms, we can use the formula for the area of a circle, A = pi * r^2, where A is the area and r is the radius. We can set up a proportion to find the radius as a function of time:

A(t) / r^2 = 10 / 10

pi * r^2 / r^2 = 10 / 8

pi = 10 / 8

r = sqrt(10 / 8) = sqrt(5/4) = (sqrt(5)/2) cm

To find the radius in meters, we need to convert the units from centimeters to meters. Since there are 100 centimeters in a meter, the radius in meters is (sqrt(5)/2) / 100 = sqrt(5) / 200 meters.

To find the number of hours it takes for the oil slick to have a radius of 1 kilometer, we need to set up another proportion:

(sqrt(5) / 200) / t = (1000 / 2) / 1

(sqrt(5) / 200) = 500

sqrt(5) = 200 * 500 = 100000

t = (sqrt(5) / 200) = 100000 / 200 = 500 hours