SOLUTION: In an right triangle prism the triangular base ABC is right angled at B,AB=5x cm and BC=12xcm.The sum of the lengths of all its edges is 18ocm. (a) Show that the volume,V cm^3, is

Algebra ->  Finance -> SOLUTION: In an right triangle prism the triangular base ABC is right angled at B,AB=5x cm and BC=12xcm.The sum of the lengths of all its edges is 18ocm. (a) Show that the volume,V cm^3, is      Log On


   

Question 1150480: In an right triangle prism the triangular base ABC is right angled at B,AB=5x cm and BC=12xcm.The sum of the lengths of all its edges is 18ocm.
(a) Show that the volume,V cm^3, is given by V=1800x^2-600x^3
(b)Find the value of x for which V has a maximum value
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


(a) Find an expression for the volume in terms of x.

The two triangular bases have side lengths 5x, 12x, and 13x.

The sum of the lengths of the sides of the two bases is 60x.

The sum of the lengths of all the edges of the prism -- the sides of the bases, plus the three edges connecting the two bases -- is 180.

Let h be the height of the prism; then

60x%2B3h+=+180
3h+=+180-60x
h+=+60-20x

The volume is the area of the base, times the height:

V+=+%28%281%2F2%29%2812x%29%285x%29%29%2860-20x%29
V+=+30x%5E2%2860-20x%29+=+1800x%5E2-600x%5E3

(b) Find the value of x that maximizes the volume. (Find the value of x for which the derivative is zero.)

dV%2Fdx+=+3600x-1800x%5E2
3600x-1800x%5E2+=+0
1800x%282-x%29+=+0
x+=+2

Note x=0 also makes the derivative zero but makes no sense in the actual problem.