2008 AMC 10B Problem 19

Attempt Problem 19 of the 2008 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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19.

A cylindrical tank with radius 44 feet and height 99 feet is lying on its side. The tank is filled with water to a depth of 22 feet. What is the volume of the water, in cubic feet?

24π−36224\pi-36\sqrt{2}

24π−24324\pi-24\sqrt{3}

36π−36336\pi-36\sqrt{3}

36π−24236\pi-24\sqrt{2}

48π−36348\pi-36\sqrt{3}

Answer: E
Concepts:cylindersectorvolume
Difficulty rating: 1680
Small Hint:

The water cross-section is a circular segment cut by a chord 22 feet below the center

Big Hint:

Its area is a 120∘120^\circ sector minus a triangle; multiply by the length 99

Solution:

The submerged cross-section is a circular segment. The chord is 4−2=24-2=2 feet below the center, and cos⁡θ=24=12,\cos\theta=\tfrac{2}{4}=\tfrac12, so the half-angle is 60∘60^\circ and the central angle is 120∘.120^\circ.

The sector area is 120360π(4)2=16π3,\tfrac{120}{360}\pi(4)^2=\tfrac{16\pi}{3}, and the triangle formed by the two radii has area 12(4)2sin⁡120∘=43.\tfrac12(4)^2\sin 120^\circ=4\sqrt3. The segment area is 16π3−43.\tfrac{16\pi}{3}-4\sqrt3.

Multiplying by the length 99 gives 9(16π3−43)=48π−363.9\left(\tfrac{16\pi}{3}-4\sqrt3\right)=48\pi-36\sqrt3.

Thus, the correct answer is E.

Problem 18#18
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