Question

a 20 ft pong water trough has ends in the form of isoceles triangles with sides that are 4ft long. determine the dimension across the top of the triangular end so that the volume of the trough is maximum.

Answer 1

Given,

Length of water trough =20ft

sides of isosceles triangle =4ft

Let's assume the height of the isosceles triangle = h and base = b

The volume of trough = area of the isosceles triangle * length of the water trough

=

In the isosceles triangle, we can write b in terms of h

The separate figure of a triangle is inserted here in fig 2.

According to that fig. and by the use of hypotenuse theorem we have,

So, volume =     

For volume to be maximum

d v / dh should be zero.

Differentiating with respect to h and applying product rule

d v/ d h =   

=

   =  

=  

=  

For,  d v/d h to be zero

   should be zero

There fore

Then the base of isosceles triangle = b=   =

For maximum volume, we got

and,