Which of the following points is not coplanar with points C, D and E? R A S U

A coin contains 9 grams of nickel and 16 grams of copper, for a total weight of 25 grams. What percentage of the metal in the co

Leya [2.2K]

X for coin Y for copper
9X+16Y=25

5 0

3 years ago

Read 2 more answers

Rewrite the following integral in spherical coordinates.

lora16 [44]

In cylindrical coordinates, we have $r^2=x^2+y^2$ , so that

$z = \pm \sqrt{2-r^2} = \pm \sqrt{2-x^2-y^2}$

correspond to the upper and lower halves of a sphere with radius $\sqrt2$ . In spherical coordinates, this sphere is $\rho=\sqrt2$ .

$1 \le r \le \sqrt2$ means our region is between two cylinders with radius 1 and $\sqrt2$ . In spherical coordinates, the inner cylinder has equation

$x^2+y^2 = 1 \implies \rho^2\cos^2(\theta) \sin^2(\phi) + \rho^2\sin^2(\theta) \sin^2(\phi) = \rho^2 \sin^2(\phi) = 1 \\\\ \implies \rho^2 = \csc^2(\phi) \\\\ \implies \rho = \csc(\phi)$

This cylinder meets the sphere when

$x^2 + y^2 + z^2 = 1 + z^2 = 2 \implies z^2 = 1 \\\\ \implies \rho^2 \cos^2(\phi) = 1 \\\\ \implies \rho^2 = \sec^2(\phi) \\\\ \implies \rho = \sec(\phi)$

which occurs at

$\csc(\phi) = \sec(\phi) \implies \tan(\phi) = 1 \implies \phi = \dfrac\pi4+n\pi$

where $n\in\Bbb Z$ . Then $\frac\pi4\le\phi\le\frac{3\pi}4$ .

The volume element transforms to

$dx\,dy\,dz = r\,dr\,d\theta\,dz = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi$

Putting everything together, we have

$\displaystyle \int_0^{2\pi} \int_1^{\sqrt2} \int_{-\sqrt{2-r^2}}^{\sqrt{2-r^2}} r \, dz \, dr \, d\theta = \boxed{\int_0^{2\pi} \int_{\pi/4}^{3\pi/4} \int_{\csc(\phi)}^{\sqrt2} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta} = \frac{4\pi}3$

4 0

2 years ago

On a very cold morning, it was -7°F. As the day went on, the temperature rose 3 degrees each hour. Which equation shows the temp

Anastasy [175]

It is C. Yep. Hope it was helpful

7 0

3 years ago

Let’s say that the father in the second couple you’re working with also has a genetic predisposition to have children of a speci

11111nata11111 [884]

Answer:

P(A) = 0.95

P(B) = 0.05

P(C) = 0.85

P(D) = 0.15

P(E) = 0.15

P(F) = 0.85

Step-by-step explanation:

6 0

3 years ago

How to find the area of a regular hexagon with a radius of 12 inches? Please help

Volgvan

$\begin{gathered} In\text{ this case, as a regular hexagon} \\ \text{radius = side} \\ Area\text{ =}3\cdot\frac{\sqrt[]{3}side^2}{2} \\ \text{side}=12in \\ side^2=144in^2 \\ Area\text{ =}3\cdot\frac{\sqrt[]{3}\cdot(144in^2)}{2} \\ \\ \text{Area}=374.1in^2 \\ \text{The regular hexagon's area is }374.1in^2 \end{gathered}$

8 0

1 year ago