What is the area of this shape?

The surface area of a right circular cone of radius r and height h is S = πr√ r 2 + h 2 , and its volume is V = 1 3 πr2h. What i

kirill115 [55]

Answer:

Required largest volume is 0.407114 unit.

Step-by-step explanation:

Given surface area of a right circular cone of radious r and height h is,

$S=\pi r\sqrt{r^2+h^2}$

and volume,

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

To find the largest volume if the surface area is S=8 (say), then applying Lagranges multipliers,

$f(r,h)=\frac{1}{3}\pi r^2 h$

subject to,

$g(r,h)=\pi r\sqrt{r^2+h^2}=8\hfill (1)$

We know for maximum volume $r\neq 0$ . So let $\lambda$ be the Lagranges multipliers be such that,

$f_r=\lambda g_r$

$\implies \frac{2}{3}\pi r h=\lambda (\pi \sqrt{r^2+h^2}+\frac{\pi r^2}{\sqrt{r^2+h^2}})$

$\implies \frac{2}{3}r h= \lambda (\sqrt{r^2+h^2}+\frac{ r^2}{\sqrt{r^2+h^2}})\hfill (2)$

And,

$f_h=\lambda g_h$

$\implies \frac{1}{3}\pi r^2=\lambda \frac{\pi rh}{\sqrt{r^2+h^2}}$

$\implies \lambda=\frac{r\sqrt{r^2+h^2}}{3h}\hfill (3)$

Substitute (3) in (2) we get,

$\frac{2}{3}rh=\frac{r\sqrt{R^2+h^2}}{3h}(\sqrt{R^2+h^2+}+\frac{r^2}{\sqrt{r^2+h^2}})$

$\implies \frac{2}{3}rh=\frac{r}{3h}(2r^2+h^2)$

$\implies h^2=2r^2$

Substitute this value in (1) we get,

$\pi r\sqrt{h^2+r^2}=8$

$\implies \pi r \sqrt{2r^2+r^2}=8$

$\implies r=\sqrt{\frac{8}{\pi\sqrt{3}}}\equiv 1.21252$

Then,

$h=\sqrt{2}(1.21252)\equiv 1.71476$

Hence largest volume,

$V=\frac{1}{3}\times \pi \times\frac{\pi}{8\sqrt{3}}\times 1.71476=0.407114$

3 0

2 years ago

The half-life of a certain radioactive material is 78 hours. An initial amount of the material has a mass of 790 kg.. Write an e

adoni [48]

<span> half life i= 78 hours
amount after 78 hours is 395 kg:
395 = 790e^(k*78)

Dividing by 790 and taking natural log
ln (395/790) = (k*78)
-0.6931 = 78k
-0.00888 = k

lets calculate how much is left after 18 hours:
Amount(18)
= 790e^(-0.00888*18)
Amount = 673.301 kg
hope this helps</span>

4 0

3 years ago

Read 2 more answers

At a farmers market, Apples are priced at 4lbs for 10$ Raigen needs 8 pounds to make Apple pie. Which equation can be used to fi

Rasek [7]

Answer:

The equation S / 8 = 2.5

3 0

3 years ago

Solve for C.<br> -5/8c=20

iris [78.8K]

-5/8c = 20....multiply both sides by -8/5 (this cancels out the -5/8 on the left
c = 20(-8/5)
c = -160/5
c = - 32

check..
-5/8(-32) = 20
160/8 = 20
20 = 20 (correct)

so c = -32

8 0

3 years ago

17. In a speech class, students must give a final speech for their exam. The speech can be 15 minutes, or within that time by 4

tekilochka [14]

Answer:

| t − 15 | = 4

Step-by-step explanation:

absolute value is weird and i see no use for it

7 0

3 years ago