A prism of an equilateral triangle base , its height is triple the length of its base if the length of its base increases by rate 0.01 cm/sec . Find the rate of increase of volume of the prism at length of base 5 root3 cm
A prism of an equilateral triangle base , its height is triple the length of its base if the length of its base increases by rate 0.01 cm/sec . Find the rate of increase of volume of the prism at length of base 5 root3 cm
I'm not sure what you mean by "the length of its base". The obvious thing is the length of a side of the triangle but another possibility is the length of an altitude.
First, can you write the formula for volume of such a prism with base side length s and height h?
the base length means -->the length of a side of the triangle
here is my attempt :
volume of prism = area of base * height .....assume that base length = x ,, height of prism = h = 3x
so
$\displaystyle v= (\frac{1}{2} \times X \times \frac{\sqrt{3}}{2}X )\times 3X$
$\displaystyle v = \frac{3 \sqrt{3}}{4} \times x^3 $by taking derivative with respect to time
so
$\displaystyle \;\frac {dv}{dt} = \frac{9 \sqrt{3}}{4} \times x^2 \times \frac{dx}{dt}$ by substituting i found that
$\displaystyle \frac {dv}{dt} = \frac{27 \sqrt{3}}{16} $
is that right?????