Question

A 25 foot long rectangular sheet of metal, 12 inches wide, is to be made into a rain gutter by turning up two sides so that they are perpendicular to the sheet. How many inches should be turned up to give the gutter its greatest capacity?

Answer 1

Answer:

The  value is  $k = 3 \ inches$

Step-by-step explanation:

From the question we are told that

   The  length is   $l = 25 \ ft$

   The  width is  $w = 12 \ inches = \frac{12}{12} = 1\ ft$

Generally let assume that (k ft)  was turned up on each side hence the remaining width is  

             $(1 - 2 k ) \ ft$

Now  the capacity is also the volume it can hold which is mathematically represented as

         $C = 25 (1- 2k) k$

          $C = 25 (k- 2k^2)$

          $C = 25k - 50k^2$

At maximum  or minimum  

     $\frac{dC}{dk} = 0$

=>    $\frac{dC}{dk} = 25 - 100k = 0$

=>    $k = 0.25\ ft$

Now  to see if the value  obtained is positive or negative we differentiate a second time

    So

        $\frac{d^2C}{dk^2} = - 100k$

at  k  =  0,25 ft

       $\frac{d^2C}{dk^2} = - 100(0.25)$

       $\frac{d^2C}{dk^2} = -25$

since a negative  value is obtained then k is  the maximum value  

converting to inches

     $k = 0.25 * 12$

     $k = 3 \ inches$