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

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 =
`(dS)/(dr)` dr +
`(dS)/(dh)` dh

`(dS)/(dr)` = (2
`\pi`r +
`\pi`
`\sqrt{h^(2)+r^(2)}` +
`\frac{\pi r^(2) }{\sqrt{h^(2)+r^(2)} }`) dr +
`(\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 +
`(18\pi )/(5)`)dr +
`(24\pi )/(5)` X 0.28.

Making dr the subject of formula , we have

(12π + 10π +
`(18\pi )/(5)`)dr = - (
`(24\pi )/(5)` X 0.28)

(
`(128\pi )/(5)`) dr = -
`(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