Question

A cone-shaped paper drinking cup is to be made to hold 24 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

Answer:

I. Radius, r = 2.90 cm

II. Height, h = 4.10 cm

Step-by-step explanation:

Given the following data;

Volume of cone = 24 cm³

To find the height and radius of the cup that will use the smallest amount of paper;

Mathematically, the volume of a cone is given by the formula;

$V = \frac{1}{3} \pi r^{2}h$  ......equation 1

Where;

V is the volume of the cone.

r is the radius of the base of the cone.

h is the height of the cone.

Substituting into the formula, we have;

$36 = \frac{1}{3} \pi r^{2}h$

Multiplying both sides by 3, we have;

$108 = \pi r^{2}h$

Making radius, r the subject of formula, we have;

$r^{2} = \frac {108}{ \pi h}$

Taking the square root of both sides, we have;

$r = \sqrt { \frac {108}{ \pi h}}$

Mathematically, the lateral surface area of a cone is given by the formula;

$LSA = \pi rl$  ......equation 2

Where;

r is the radius of a cone

l is the slant height of a cone.

To find the slant height, we would apply the Pythagorean' theorem;

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

Substituting r into the above equation, we have;

$l = \sqrt {\frac {108}{\pi h} + h^{2}}$

Substituting the values of r and l into eqn 2, we have;

$LSA = \pi * \sqrt { \frac {108}{ \pi h}} * \sqrt {\frac {108}{\pi h} + h^{2}}$

Simplifying further, we have;

$LSA = \sqrt {108} * \sqrt { \frac {\pi h^{3} + 108}{\pi h}}$

$LSA = \sqrt {108} * \sqrt { \frac {108}{ h^{2}} + \pi h}}$

Next, to find the value of h, we differentiate the above mathematical equation with respect to h;

$\frac {dS}{dh} = \sqrt {108} * (\pi - \frac {216}{h^{3}}) * (\pi h + \frac {108}{h^{2}})$

Limiting $\frac {dS}{dh}$ w.r.t 0;

$\frac {dS}{dh} = 0$

$(\pi - \frac {216}{h^{3}}) = 0$

Rearranging the equation, we have;

$\pi = \frac {216}{h^{3}}$

We know that π = 3.142

$3.142 = \frac {216}{h^{3}}$

Cross-multiplying, we have;

$3.142h^{3} = 216$

$h^{3} = \frac {216}{3.142}$

$h^{3} = 68.75$

Taking the cube root of both sides, we have;

Height, h = 4.10 cm

Lastly, we find the value of r;

$r = \sqrt { \frac {108}{ \pi h}}$

$r = \sqrt { \frac {108}{3.142 * 4.10}}$

$r = \sqrt { \frac {108}{12.88}}$

$r = \sqrt {8.39}$

Radius, r = 2.90 cm

Answer 2

The height and radius of the cup that will use the smallest amount of paper is;

Radius = 2.52 cm

Radius = 2.52 cmHeight = 3.58 cm

Let us first state some relevant formulas;

Volume of a cone is;

V = ⅓πr²h

Surface area of a cone is;

S = πrL

Where L is Slant height and has a formula;

L = √(h² + r²)

We are told that the cone is to hold 24 cm³. Thus; V = 24 cm³

24 = ⅓πr²h

πr²h = 72

r = √(72/πh)

Putting √(72/πh) for r in the Slant height equation gives;

L = √(h² + (72/πh))

Thus;

S = π × √(72/πh) × √(h² + (72/πh))

Differentiating with respect to h gives;

dS/dh = √72 × (π - 144/h³) × 1/√(πh + 72/h²)

At dS/dh = 0,we will have;

(π - 144/h³) = 0

Thus;

h³ = 144/π

h = 3.58 cm

Thus, from r = √(72/πh);

r = √(72/(π × 3.58))

r = 2.52 cm

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