Question

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 28/100. 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.

Answer 1

Answer: r ≈ 5.95

Step-by-step explanation:

Let S be the surface area , then

S = $\pi$$r^{2}$ + $\pi$r$\sqrt{r^{2}+h^{2}}$

Differentiating S with respect to r , we have

dS = $\frac{dS}{dr}$ dr + $\frac{dS}{dh}$ dh

$\frac{dS}{dr}$ = (2$\pi$r +$\pi$$\sqrt{h^{2}+r^{2}}$ + $\frac{\pi r^{2} }{\sqrt{h^{2}+r^{2}} }$) dr + $\frac{\pi rh }{h^{2}+r^{2}  }$ dh

For the area to remain the same it means dS = 0

since r = 6cm , h = 8cm , dh = 0.28 then we have

0 = (12π + 10π +$\frac{18\pi }{5}$)dr + $\frac{24\pi }{5}$ X 0.28.

Making dr the subject of formula , we have

(12π + 10π +$\frac{18\pi }{5}$)dr = - ($\frac{24\pi }{5}$ X 0.28)

($\frac{128\pi }{5}$) dr = - $\frac{6.72\pi }{5}$

(640$\pi$)dr = - 33.6$\pi$

dr = - 33.6$\pi$ / 640$\pi$

dr = 0.05

This means that the radius should decrease by 0.05 for the area to remain the same. That means the radius = 6 - 0.05

r ≈ 5.95