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Diagram
A frustum is formed by removing the top section of a right pyramid of rectangular base EFGH as shown in the diagram below. It is given that AE=BF=CG=DH=OA=12cm, FG=20cm and EF=30cm.
(i)The vertical height of the frustum is labeled h cm. Show that the value of h is 7.9215, correct to 5 significant figures.
(ii)Hence, calculate the volume of the frustum, correct to the nearest whole number.

Draw a line from the point O down to the center of the bottom.
Label the point where that line cross rectangle ABCD as X.
Label the point where that line hits the bottom in the center of rectangle EFGH as Y.
Let [AB] be known as, "the length of AB."  This applies to any
This makes h be [XY].

It is known that triangles EYO and GYO are reflections of one another.
It can also be seen that they are right triangles.

It can be seen that EG is the hypotenuse of the right triangle EGF.
From this, it can be seen that the length of EG is sqrt([EF]^2 + [FG]^2).

It can be seen that [EY] = [EG]/2.
It can be noted that EYO is a right triangle with the hypotenuse EO and lower side EY.
Since, for right triangles, [EY]^2 + [YO]^2 = [EO]^2, and we know [EY] and [EO],
this will let us find [YO] as [YO] = sqrt([EO]^2 - [EY]^2).

It can be seen that [EO] = [EA] + [AO] = 12 + 12 = 24.
It can be seen that [EY] = [EG]/2 = sqrt([EF]^2 + [FG]^2)/2 = sqrt(900+400)/2 = sqrt(1300)/2.
It can be seen that sqrt(1300)/2 = sqrt(1300/4) = sqrt(325).
That makes [YO] = sqrt(576 - 325) = sqrt(251).
Since [XY] is half ot this, we know that [XY] = sqrt(251)/2 = 7.9215.

By the way, I keep all values in a spreadsheet and don't round until the end.
It is really 7.921489759.

The area of EFGH is 20*30 = 600.
The area of ABCD is (20/2)(30/2) = 150.

The volume of a cone is the area of the base times the height over 3.
We have two cones, both with top O.
The volume of the bigger is [EF][FG][YO]/3, which is 30*20*sqrt(251)/3 = 3168.5959.
The volume of the smaller is [AB][BC][XO]/3, which is 15*10*[sqrt(251)/2]/3 = 396.07449.

The volume of the shaded figure is the difference, and that is 2772.5.

It should also be noted that the smaller volume is 1/8 of the larger since
each side is half as long.

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#### Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.