BTGModeratorVI wrote: ↑Tue Dec 15, 2020 6:58 am
A tank is filled with gasoline to a depth of exactly 2 feet. The tank is a cylinder resting horizontally on its side, with its circular ends oriented vertically. The inside of the tank is exactly 6 feet long. What is the volume of gasoline in the tank?
(1) The inside of the tank is exactly 4 feet in diameter.
(2) The top surface of the gasoline forms a rectangle that has an area of 24 square feet.
Answer:
D
Source: official guide
Solution:
Question Stem Analysis:
We need to determine the volume of gasoline in the tank, given that the tank is a cylinder resting horizontally on its side, with its circular ends oriented vertically. The inside of the tank is exactly 6 feet long, and the tank is filled with gasoline to a depth of exactly 2 feet.
Notice that when the tank is resting vertically with the circular base on the ground (like an upright soda can), the height of the tank is 6 feet.
Statement One Alone:
Since the diameter of the tank is 4 feet, which is exactly twice the depth of 2 feet currently of the gasoline filled inside the tank, we see that the tank is actually half full. Since the diameter of the tank is 4 feet, its radius is 2 feet. Since in the question stem analysis, we determined the height of the tank is 6 feet, we can determine the full capacity of the tank. Since the gasoline is filled to half capacity, dividing the full capacity of the tank by 2 will yield the volume of the gasoline in the tank. Statement one alone is sufficient.
Statement Two Alone:
We see that the rectangle (i.e, the top surface of the gasoline) has a width of 4 feet since the area is 24 square feet and its length is 6 feet. The width of the rectangle is actually a chord of the circular base of the tank. Let the distance between the center of the circle and this chord be x. Recall that we are told that the depth of the gasoline is 2 feet, therefore, the radius of the circular base is x + 2 if the center is above the gasoline level and 2 - x if the center is below the gasoline level. In the former case, a right triangle is formed where the legs have length 2 and x, and where the hypotenuse (which is the radius) has length x + 2. Using the Pythagorean theorem, we obtain:
(x + 2)^2 = x^2 + 2^2
x^2 + 4x + 4 = x^2 + 4
4x = 0
x = 0
In this case, we see that the distance between the center and the gasoline level is 0, which means that the tank is actually half full and the diameter of the circular ends is 4.
Assuming the level of the gasoline is above the center, we can proceed as above and solve (2 - x)^2 = x^2 + 2^2, which also yields x = 0. Once again, the tank is half full and the diameter is 4.
We see that in either case, we have enough information to determine the volume of gasoline in the tank.
Answer: D
