Question

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.)

Answer 1

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,

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

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

Learn more:

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2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Mensuration

Keywords: cone shaped paper, drinking cup, volume, $36{\text{ c}}{{\text{m}}^3}$, height of cone, cup, smallest amount of paper, water.

Answer 2

The formula for volume of cone is:

V = π r^2 h / 3

or

π r^2 h / 3 = 36 cm^3

Simplfying in terms of r:

r^2 = 108 / π h

To find for the smallest amount of paper that can create this cone, we call for the formula for the surface area of cone:

S = π r sqrt (h^2 + r^2)

S = π sqrt(108 / π h) * sqrt(h^2 + 108 / π h) 

S = π sqrt(108 / π h) * sqrt[(π h^3 + 108) / π h] 

Surface area = sqrt (108) * sqrt[(π h + 108 / h^2)] 

Getting the 1st derivative dS / dh then equating to 0 to get the maxima value:

dS/dh = sqrt (108) ((π – 216 / h^3) * [(π h + 108/h^2)^-1/2] 

Let dS/dh = 0 so,

 π – 216 / h^3 = 0 

h^3 = 216 / π

h = 4.10 cm

Calculating for r:

r^2 = 108 / π (4.10)

r = 2.90 cm

 

Answers:

 h = 4.10 cm

r = 2.90 cm