Finding the volume of the pyramid and also determining the increase and decrease of its volumeThe volume of a pyramid with a square base x units on a side and a height of h is V = 1 3 x 2 h.1) Suppose x = e t and h = e − 2 t . Use the chain rule to find V 1 ( t ).2) Does the volume of the pyramid increase or decrease as t increases.

Question

Finding the volume of the pyramid and also determining the increase and decrease of its volumeThe volume of a pyramid with a square base x units on a side and a height of h is   V  =            1      3            x    2    h.1) Suppose   x  =      e    t   and   h  =      e          −      2      t      . Use the chain rule to find       V    1    (  t  ).2) Does the volume of the pyramid increase or decrease as t increases.

furniranizq · Accepted Answer

Step 1  V  =      1    3        e          2      t            e          −      2      t        =      1    3        e          2      t      −      2      t        =      1    3        e          0        =      1    3  Thus       V    ′    (  t  )  =  0. But if you insist on using the chain rule then, we apply the product rule and then the chain ruleStep 2      V    ′    =  −      2    3        e          2      t            e          −      2      t        +      2    3        e          2      t            e          −      2      t        =  0Clearly the volume of the pyramid clearly is a constant so doesn&#039;t increase or decrease.