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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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