The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. The ship uses 30 barrels of oil in traveling 85 miles at 42 mi/h. How many barrels of oil are used when the ship travels 26 miles at 54 mi/h? Round your answer to the nearest tenth of a barrel, if necessary.
Let:
- x be the amount of oil used
- d be the distance traveled in miles
- s be the speed in mi/h
We are given that the amount of oil used varies jointly with the distance and the square of the speed, so we can write the equation as:
x = kds^2
where k is the constant of variation.
From the first scenario where the ship uses 30 barrels of oil in traveling 85 miles at 42 mi/h, we have:
30 = kd(85)^2(42)^2
30 = k(85)(1764)
30 = 150540k
k = 30 / 150540
k ≈ 0.000199
Now we can use the constant k to find how many barrels of oil are used when the ship travels 26 miles at 54 mi/h:
x = 0.000199(26)(54)^2
x = 0.000199(26)(2916)
x ≈ 1.429 barrels
Therefore, when the ship travels 26 miles at 54 mi/h, approximately 1.4 barrels of oil are used.