Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Problem number 29. Two identical cylindrical water tanks contain the same amount of water. One cylinder is standing upright and the other is leaning  against it as can be seen in the bottom left and the water level in each of them is the same as in the picture. The bottom of each of the cylinders is a circle  with area 3 pi meters square. How much water does each tank contain? Alright so for this problem let's first label some stuff just so that we know what we're  referring to. The height of the original tank is H, the radius is R, and the slant height we're going to call L. The first thing we notice is that actually  if we take the slanted tank and put it back upright, this height right here, this tank right here, has a volume half of what it would have had if it was just  filled normally with height L and radius R and this actually just comes from observing this diagram. So rotating that leaning cylinder to the vertical  position, we know that the volume of the original tank is pi R squared H and the second tank would be pi R squared L but we divide by 2 because it's half of  that and that gives us that L is equal to 2H. Then the right triangle with leg H and hypotenuse L is actually a 30-60-90 right triangle because L over H  gives us the ratio 2 and whenever you have that hypotenuse over one of the legs is equal to 2 you have a 30-60-90 right triangle. Then the right triangles  with legs L and 2R is also a 30-60-90 degree triangle. Since the base circle of the cylinder has an area of 3pi, we know that R is equal to square root of 3.  From the 30-60-90 triangle relations which are the smallest leg is equal to R, the second leg is equal to R square root of 3 and the hypotenuse is  equal to 2R, we know that L is equal to 2R square root of 3 which is equal to 6 and H is equal to 1 half L equals 3. Thus each cylinder contains 3pi H or 3pi  times 3 which is equal to 9pi meters cubed of water and our correct answer is C.

Video Summary

In this problem, two identical cylindrical tanks contain equal water amounts. Given each cylinder's base area is \(3\pi\) square meters, calculations reveal the upright tank's height (\(H\)) is 3 meters, while the slanted tank's slant height (\(L\)) is 6 meters. Using geometry, specifically 30-60-90 triangle relationships, \(L\) equals twice \(H\). The tank's base circle has radius \(R = \sqrt{3}\). Each tank's water volume is \(3\pi \cdot H = 9\pi\) cubic meters, making the correct answer \(C\), as both tanks contain 9\pi cubic meters of water each.