SOLUTION: A frustum of a pyramid consist of a square of length 10cm and a top square of length 7cm.the height of the frustum is 6cm.calculate to the nearest whole number (1)surface area (2)v

Algebra ->  Surface-area -> SOLUTION: A frustum of a pyramid consist of a square of length 10cm and a top square of length 7cm.the height of the frustum is 6cm.calculate to the nearest whole number (1)surface area (2)v      Log On


   

Question 1141887: A frustum of a pyramid consist of a square of length 10cm and a top square of length 7cm.the height of the frustum is 6cm.calculate to the nearest whole number (1)surface area (2)volume
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


Picture the frustum as a larger pyramid with a smaller similar pyramid cut off the top.

(1) Surface area

Let x be the height of the smaller pyramid; then the height of the larger pyramid is x+6.

The large and small pyramids are similar figures, so the ratios of corresponding lengths are equal. So

x%2F%28x%2B6%29+=+7%2F10
7x%2B42+=+10x
42+=+3x
x+=+14

So the height of the large pyramid is x+6 = 20.

The slant height of the large pyramid can then be found using the Pythagorean Theorem: the slant height is the hypotenuse of a right triangle with the height of the large pyramid as one leg and half the base of the large pyramid as the other leg.

slant height of large pyramid = sqrt%2820%5E2%2B5%5E2%29+=+sqrt%28425%29+=+5%2Asqrt%2817%29

Then by the similarity of the two pyramids, the slant height of the frustum is 3/10 of the slant height of the large pyramid.

slant height of frustum = %283%2F10%29%285%2Asqrt%2817%29%29+=+%283%2F2%29sqrt%2817%29

Finally, the surface area of the frustum is the area of the two bases, plus the area of four congruent trapezoids, each with bases 10 and 7 and height (3/2)sqrt(17).

You can do the calculations....

(2) Volume

The volume of a pyramid is one-third base times height:

volume of large pyramid: %281%2F3%29%2810%5E2%29%2820%29
volume of small pyramid: %281%2F3%29%287%5E2%29%2814%29
volume of frustum: volume of large pyramid minus volume of small pyramid

Again you can do the calculations.

There is also a concise formula for the volume of a frustum of a pyramid with bases a and b and height h:

%281%2F3%29%28a%5E2%2Bab%2Bb%5E2%29%28h%29

You can get some good practice (and maybe learn something new) by finding the volume by both methods and seeing that the results are the same.