A cylinder and a cone have a common base and height. Calculate the volume of the cylinder if the volume of the cone = 10.

equipokypip1
2022-09-13
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asked 2022-09-15

Finding the volume of an object with 3 parameters

I know how to find the volume of a sphere/ball around x-axis using:

$V=\pi {\int}_{a}^{b}{f}^{2}(x)dx$

Lets say if:

${x}^{2}+{y}^{2}={r}^{2}$

We do: $y=\sqrt{{r}^{2}-{x}^{2}}$

So now: $V=\pi {\int}_{-r}^{r}(\sqrt{{r}^{2}-{x}^{2}}{)}^{2}dx$

But the problem starts here:

Im given an equation of ellipsoid, it has 3 parameters:

${x}^{2}+4{y}^{2}+4{z}^{2}\le 4$

What do i do with the z parameter? How do i build y now? how does it fit to the equation by integrals of V?

I would like an explanation and not just a solution - because its homework.

I know how to find the volume of a sphere/ball around x-axis using:

$V=\pi {\int}_{a}^{b}{f}^{2}(x)dx$

Lets say if:

${x}^{2}+{y}^{2}={r}^{2}$

We do: $y=\sqrt{{r}^{2}-{x}^{2}}$

So now: $V=\pi {\int}_{-r}^{r}(\sqrt{{r}^{2}-{x}^{2}}{)}^{2}dx$

But the problem starts here:

Im given an equation of ellipsoid, it has 3 parameters:

${x}^{2}+4{y}^{2}+4{z}^{2}\le 4$

What do i do with the z parameter? How do i build y now? how does it fit to the equation by integrals of V?

I would like an explanation and not just a solution - because its homework.

asked 2022-08-27

So, the volume of the solid of revolution is

Volume = $V=\pi {\int}_{a}^{b}(9-{x}^{2}{)}^{2}dx.$

To find the volume, integrate from x =??? to x = 3

Volume = $V=\pi {\int}_{a}^{b}(9-{x}^{2}{)}^{2}dx.$

To find the volume, integrate from x =??? to x = 3

asked 2022-08-08

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of th volume when r = 24 inches.

asked 2022-07-18

Finding a rate of sphere area increase given its volume increase

Volume of sphere, $V={\displaystyle \frac{4}{3}}\pi {r}^{3}$

Surface area of sphere $S=4\pi {r}^{2}$.

If we know, $\frac{dV}{dt}}=R$.

Let us consider both volume and area as composite functions, thus $\frac{dV}{dt}}={\displaystyle \frac{dV}{dr}}\times {\displaystyle \frac{dr}{dt}}=4\pi {r}^{2}\times {\displaystyle \frac{dr}{dt}}=R$

whence $\frac{dr}{dt}}={\displaystyle \frac{R}{4\pi {r}^{2}}$ since $\frac{dS}{dt}}=6\pi r{\displaystyle \frac{dr}{dt}$, let the value of the $\frac{dr}{dt}$ into the second equation, to get the answer. Is this approach logically correct?

Volume of sphere, $V={\displaystyle \frac{4}{3}}\pi {r}^{3}$

Surface area of sphere $S=4\pi {r}^{2}$.

If we know, $\frac{dV}{dt}}=R$.

Let us consider both volume and area as composite functions, thus $\frac{dV}{dt}}={\displaystyle \frac{dV}{dr}}\times {\displaystyle \frac{dr}{dt}}=4\pi {r}^{2}\times {\displaystyle \frac{dr}{dt}}=R$

whence $\frac{dr}{dt}}={\displaystyle \frac{R}{4\pi {r}^{2}}$ since $\frac{dS}{dt}}=6\pi r{\displaystyle \frac{dr}{dt}$, let the value of the $\frac{dr}{dt}$ into the second equation, to get the answer. Is this approach logically correct?

asked 2022-08-19

Finding the volume of liquid in an inclined cylinder

A right circular cylinder is at an incline of ${15}^{\circ}$ from the horizontal and the liquid is level with the lowest point of the top rim of the can. The radius is 3.2004 cm and the height is 11.938 cm. What is the volume of the liquid?

I believe I should use integration with cross-sections of rectangles. The width of each rectangle would be the diameter of the cylinder. I believe my limits of integration would be from 0 to 4.7 \cdot \sin(15^{\circ}) or 1.2164. I'm not sure how to figure out the changing lengths of the rectangles.

Am I on the right track?

${\int}_{0}^{4.7\mathrm{sin}(15\xb0)}6.4008?dy$

A right circular cylinder is at an incline of ${15}^{\circ}$ from the horizontal and the liquid is level with the lowest point of the top rim of the can. The radius is 3.2004 cm and the height is 11.938 cm. What is the volume of the liquid?

I believe I should use integration with cross-sections of rectangles. The width of each rectangle would be the diameter of the cylinder. I believe my limits of integration would be from 0 to 4.7 \cdot \sin(15^{\circ}) or 1.2164. I'm not sure how to figure out the changing lengths of the rectangles.

Am I on the right track?

${\int}_{0}^{4.7\mathrm{sin}(15\xb0)}6.4008?dy$

asked 2022-09-25

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly $m\le n$. In general, for computational purposes, what I have managed to do is define volume as the product of absolute values of vectors obtained by gram-schmidt orthogonalizationn. (Makes sense right? That's the natural interpretation when we say volume)

I had to do two things, firstly to show that this definition of volume is a well defined one (i.e. any set of orthogonal vectors obtained by the process will give the same volume), and secondly to find a quick way to do this. I managed to prove the first one by induction, but the second part is a little bit of a problem. I managed to obtain formulae for small dimensions as 2,3 or even 4 but this process is impractical for any bigger dimensions as the substitutions for smaller dimensions into the formula for the next dimension becomes exponentially complicated

How does one prove that the gram determinant is equal to the volume of a paralleletope spanned by a set of vectors?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly $m\le n$. In general, for computational purposes, what I have managed to do is define volume as the product of absolute values of vectors obtained by gram-schmidt orthogonalizationn. (Makes sense right? That's the natural interpretation when we say volume)

I had to do two things, firstly to show that this definition of volume is a well defined one (i.e. any set of orthogonal vectors obtained by the process will give the same volume), and secondly to find a quick way to do this. I managed to prove the first one by induction, but the second part is a little bit of a problem. I managed to obtain formulae for small dimensions as 2,3 or even 4 but this process is impractical for any bigger dimensions as the substitutions for smaller dimensions into the formula for the next dimension becomes exponentially complicated

How does one prove that the gram determinant is equal to the volume of a paralleletope spanned by a set of vectors?

asked 2022-08-20

Help with finding the volume of a solid of revolution

Revolve ${x}^{2}+4{y}^{2}=4$.

a. About $y=2$.

b. About $x=2$.

The answer at the back of the book is the same for both

$a.2\pi {\int}_{0}^{2}8\sqrt{1-\frac{{x}^{2}}{4}}dx=78.95684$

$b.2\pi {\int}_{0}^{1}8\sqrt{4-4{y}^{2}}dy=78.95684$

I tried splitting the ellipse into its upper and lower half but I still couldn't get the answer. I get that its symmetrical that's why it's multiplied to two but I don't know how you get the equation inside the integral.

Revolve ${x}^{2}+4{y}^{2}=4$.

a. About $y=2$.

b. About $x=2$.

The answer at the back of the book is the same for both

$a.2\pi {\int}_{0}^{2}8\sqrt{1-\frac{{x}^{2}}{4}}dx=78.95684$

$b.2\pi {\int}_{0}^{1}8\sqrt{4-4{y}^{2}}dy=78.95684$

I tried splitting the ellipse into its upper and lower half but I still couldn't get the answer. I get that its symmetrical that's why it's multiplied to two but I don't know how you get the equation inside the integral.