3 consecutive integers is 183?

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

What is - 6<br><br> 5p - p when you simplify it

Darina [25.2K]

5p-p=4p. The question doesn't make sense because ther is no sign after the minus 6. If it is an X it would be -24p. If it is an add it would be -6+4p. If it was a - it would be -6-4p.

6 0

3 years ago

What is 8 2/5% of 75? Write your answer as a decimal

likoan [24]

Answer: 0.875

Step-by-step explanation:

The answer that has already been given is perfect in and by itself. I just want to give you some methods for doing these type of questions so that you won’t stagger when you see these type of questions in exams (like the SAT).

Whenever you see such questions, remember the following things. These are conversions of English words to Mathematical tools.

What

=

Any variable like

x

Percent / Percentage / %

=

Divide by 100

of

=

Multiply

is

=

Put an equal to (

=

) sign

I’ll give you an example through a question.

What percent of 200 is 5?

Ans: Replace English words by mathematical tools according to the rules I told you.

x

( What) /100 ( percent)

×

( of) 200

=

( is) 5

In a refined form,

x

100

×

200

=

5

Calculate x. On calculating, x comes out to be

2.5

So, 2.5 percent of 200 is 5.

Now your question is actually very simple if you follow my rules.

You want 87.5%. By the rules, % means divided by 100. So

87.5

%

=

87.5

100

and now using the same strategy, Ans:

875

1000

Disregard the next section if you know how to convert a decimal to a fraction

Section Starts

First, I’ll write the decimal form. Whenever you divide by a number that is in form of 1 followed by

n

number of zeroes. (here 1 is followed by 2 zeroes), you should shift the decimal point to the left side by n places.

In this case, shift decimal point in 87.5 to the left by 2 places. So this becomes 0.875.

This is the answer in decimal point.

Now simultaneously, I’ll teach you how to convert a decimal into a fraction.

Here, you see 2 types of “numbers”. One to the left side of the decimal point and one to the right side of the decimal point.

Here, left side of decimal point is 0 and right side of decimal point is 875.

Count the number of digits in the number to the right side of the decimal point (Here, the number of digits is 3)

Remove the decimal point and divide the number by a number formed by 1 followed by the “digits” we obtained (i.e. 3).

So, 875/1000. Simplify this, and you get your answer.

5 0

2 years ago

hello! i struggle with math A LOT so if u guys can help me with these questions with the work i would appreciate a lot

elena-s [515]

Pretty sure 1 is 6/8

4 0

3 years ago

Read 2 more answers

ANSWER THIS U GET BRAINLIEST:)

LenaWriter [7]

Answer:

B

Step-by-step explanation:

3 0

3 years ago

Read 2 more answers