linear approximation and derivative help

Question

A right circular cone of height h and base radius r has total surface area S consisting of its base area plus its side area, leading to the formula:

S= pi*r^2+pi*r*sqrt(r^2+h^2)

Suppose you start out with a cone of height 8 cm and base radius 6 cm, and you want to change the dimensions in such a way that the total surface area remains the same. Suppose you increase the height by 17/100. In this problem, use tangent line approximation to estimate the new value of r so that the new cone has the same total surface area.

estimate the value of r.
I have no idea what to do with this problem. I know i should use the are formulas and the derivatives but am unsure how tow to go about it.

Answer 1

The most straightforward way to approach this problem is to think of r as being a function of h such that the area of the cone stays unchanged.  This is a constraint condition.

 

An expression for the derivative of this function with respect to h   (i.e  d r(h) / dh  ) can be obtained by the method of implicit differentiation.    One differentiates  S with respect to h.   Under the constraint condition, this derivative will be zero.  The chain rule is used several times in working out the derivative.  Use of the chain rule produces dr(h)/dh in several places.    After some rearrangement, this procedure leads to:

 

   (dr(h) / dh) [ 2 r + sqrt(r² + h²) + r²/sqrt(r² + h²) ]  = - r h/ sqrt(r² + h²)

 

 To leading order (spirit of the tangent line approximation) one can substitute r = 6 and h =8 into this equation to get a value for d r(h) /dh.  I find  dr(h) /dh = - .233

 

  Thus as h  increases from 8 to 8.17,   r will decrease by  (.233 ) .17  or .0396  .  

  So  the new value of r is  6 - .0396 = 5..96

 

This last part assumes that .17 is the additive amount by which h increases,  not a percent change.