Alyce Wilkinson
2021-01-08
Answered

Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate (in cubic meters per second) through the upper hemisphere $(z>0)\text{}\text{of the sphere}\text{}{x}^{2}+{y}^{2}+{z}^{2}=1$ .

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Szeteib

Answered 2021-01-09
Author has **102** answers

Similar to circle sphere is a two dimensional space where the set of points that are at the same distance r from a given point in a three dimensional space. In analytical geometry with a center and radius is the locus of all points is called sphere
Given:
The upper hemisphere of the sphere is ${x}^{2}+{y}^{2}+{z}^{2}=1$
Formula used:
$\text{Volume}=\int \int \int (\xf7F)dv$

$V=(0,0,z)$

$\xf7F=\frac{d}{dx}(0)+\frac{d}{dy}(0)+\frac{d}{dz}(z)$
On solving the value,
$\xf7F=0+0+1=1$

$\text{Volume}=\int \int \int (\xf7F)dv$
Where,
$dv=\frac{4}{3}\pi {r}^{3}$

${x}^{2}+{y}^{2}+{z}^{2}=1={r}^{2}$
Where, r = 1
$\frac{4}{3}\pi {r}^{3}$
Substituting the value of r,
then the required value is $\frac{4\pi}{3}$

asked 2020-12-15

The area of a rectangular cloth is

asked 2022-07-15

Finding formula for parabolic cone with constant ratio of volume to surface area of each circular cross section

I'm trying to express an equation for a parabolic cone (paraboloid?) where the ratio of volume to surface area of the circle through covering every cross section throughout the height of the 'cone' is constant. Presumably there is some parabola equation that creates a line that if spun around to create a parabolic cone shape would give this outcome.

I'm trying to express an equation for a parabolic cone (paraboloid?) where the ratio of volume to surface area of the circle through covering every cross section throughout the height of the 'cone' is constant. Presumably there is some parabola equation that creates a line that if spun around to create a parabolic cone shape would give this outcome.

asked 2022-03-17

Assume T: $R\wedge m$ to $R\wedge n$ is a matrix transformation with matrix A.

Prove that if the columns of A are linearly independent then T is one to one. (i.e injective) (Hint: Remember the matrix transformations satisfy the linearity properties.)

Linearity Properties:

If A is a matrix, v and w are vectors and c is a scalar then,

A0=0

A(cv)=cAv

Prove that if the columns of A are linearly independent then T is one to one. (i.e injective) (Hint: Remember the matrix transformations satisfy the linearity properties.)

Linearity Properties:

If A is a matrix, v and w are vectors and c is a scalar then,

A0=0

A(cv)=cAv

asked 2022-05-09

Let $n$ points be placed uniformly at random on the boundary of a circle of circumference 1.

These $n$ points divide the circle into $n$ arcs.

Let ${Z}_{i}$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.

What is $E[X]$ and $Var[X]$?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)

These $n$ points divide the circle into $n$ arcs.

Let ${Z}_{i}$ for $1\le i\le n$ be the length of these arcs in some arbitrary order, and let $X$ be the number of ${Z}_{i}$ that are at least $\frac{1}{n}$.

What is $E[X]$ and $Var[X]$?

Any hints will be appreciated. Thanks..

(By the way this problem is exercise 8.12 from the book 'Probability and Computing' by Mitzenmacher and Upfal)

asked 2021-11-09

(a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt.

(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30 m?

asked 2022-07-21

Find the Locus of the Orthocenter

Vertices of a variable triangle are $(3,4)\phantom{\rule{0ex}{0ex}}(5\mathrm{cos}\theta ,5\mathrm{sin}\theta )\phantom{\rule{0ex}{0ex}}(5\mathrm{sin}\theta ,-5\mathrm{cos}\theta )$ where $\theta \in \mathbb{R}$. Given that the orthocenter of this triangle traces a conic, evaluate its eccentricity.

I was able to find the locus after three long pages of cumbersome calculation. I found the equations of two altitudes of this variable triangle using point slope form of equation of a straight and then solved the two lines to get the orthocenter. However, the equation turned out to be of a non standard conic. I evaluated its Δ to find that it's an ellipse, but I don't know how to find the eccentricity of a general ellipse.

Moreover, there must be a more elegant way of doing this since the questions in my worksheet are to be solved within 5 to 6 minutes each but this took way long using my approach.

Vertices of a variable triangle are $(3,4)\phantom{\rule{0ex}{0ex}}(5\mathrm{cos}\theta ,5\mathrm{sin}\theta )\phantom{\rule{0ex}{0ex}}(5\mathrm{sin}\theta ,-5\mathrm{cos}\theta )$ where $\theta \in \mathbb{R}$. Given that the orthocenter of this triangle traces a conic, evaluate its eccentricity.

I was able to find the locus after three long pages of cumbersome calculation. I found the equations of two altitudes of this variable triangle using point slope form of equation of a straight and then solved the two lines to get the orthocenter. However, the equation turned out to be of a non standard conic. I evaluated its Δ to find that it's an ellipse, but I don't know how to find the eccentricity of a general ellipse.

Moreover, there must be a more elegant way of doing this since the questions in my worksheet are to be solved within 5 to 6 minutes each but this took way long using my approach.

asked 2022-06-24

A circle has diameter AD of length 400.

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?