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joejoefofana

18.09.2019 •

Mathematics

Use triple scalar products to find volume of the parallelepiped having adjacent edges
u = i+j+3k, v = 3j +3k, and w = 3i + 3k.

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Ответ:

ZVSaniya0420

15.03.2023

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

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