A cone-shaped paper drinking cup is to be made to hold 36 cm3 of water. find the height and radius of the cup that will use the

Question

mote1985 [20]

3 years ago

5

A cone-shaped paper drinking cup is to be made to hold 36 cm3 of water. find the height and radius of the cup that will use the

smallest amount of paper. (round your answers to two decimal places.)

Mathematics

2 answers:

HACTEHA [7] · Answer 1 · 2022-04-09T07:35:45+03:00

The height of the cone is $\boxed{4.10}$ and the radius of the cone is $\boxed{2.90}.$

Further explanation:

The volume of the cone is $\boxed{V = \dfrac{1}{3}\left( {\pi {r^2}h} \right)}.$

The surface area of the cone is $\boxed{S=\pi \times r\times l}$

Here l is the slant height of the cone.

The value of the slant height can be obtained as,

$\boxed{l = \sqrt {{h^2} + {r^2}} }$ .

Given:

The volume of the cone shaped paper drinking cup is $36{\text{ c}}{{\text{m}}^3}$ .

Explanation:

The volume of the cone shaped paper drinking cup $36{\text{ c}}{{\text{m}}^3}$ .

$\begin{aligned}V&=36\\\frac{1}{3}\left({\pi {r^2}h}\right) &= 36\\{r^2}&= \frac{{108}}{{\pi h}}\\\end{aligned}$

The surface area of the cone is,

$\begin{aligned}S &= \pi\times\sqrt {\frac{{108}}{{\pi h}}}\times\sqrt {{h^2} + \frac{{108}}{{\pi h}}}\\&= \sqrt{108}\times\sqrt{\frac{{\pi {h^3} + 108}}{{\pi h}}}\\&=\sqrt {108}\times\sqrt {\pi h + \frac{{108}}{{{h^2}}}}\\\end{aligned}$

Differentiate above equation with respect to h.

$\dfrac{{dS}}{{dh}}=\sqrt {108}\times \left( {\pi - \dfrac{{216}}{{{h^3}}}}\right)\times {\left( {\pi h + \dfrac{{108}}{{{h^2}}}}\right)^{ - \dfrac{1}{2}}}$

Substitute 0 for $\dfrac{{dS}}{{dh}}$ .

$\begin{aligned}\pi- \dfrac{{216}}{{{h^3}}}&= 0\\\dfrac{{216}}{{{h^3}}}&= \pi\\\dfrac{{216}}{{3.14}} &= {h^3}\\h &= 4.10\\\end{aligned}$

The radius of the cone can be obtained as,

$\begin{aligned}{r^2}&=\frac{{108}}{{\pi \left({4.10} \right)}}\\{r^2}&= \frac{{108}}{{3.14 \times 4.10}}\\{r^2}&= 8.40\\r&= \sqrt {8.40}\\r &= 2.90\\\end{aligned}$