Question

(Old Final Exam Problem-A2011) 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= π r2 + π r r2+h2 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 15/100. 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. The estimated value of r =

Answer 1

Answer:

  5.972 cm

Step-by-step explanation:

Using implicit differentiation, we have ...

  0 = (2πr +π(√(r^2+h^2) +r^2/√(r^2+h^2))·dr/dh +πrh/√(r^2+h^2)

For the given values of r and h, this is ...

  0 = (12π +π(10 +36/10))dr/dh +48π/10

  dr/dh = -4.8/(12+13.6) = -0.1875

Then the tangent line is ...

  r = -0.1875(h -8) +6

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For a height of 8 +15/100 = 8.15, the estimated value of r is ...

  r = -0.1875(8.15 -8) +6 = -0.1875(0.15) +6 = 5.971875

The estimated value of r is 5.972 cm.