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saw5 [17]
3 years ago
5

Choose the best coordinate system to find the volume of the portion of the solid sphere rho <_4 that lies between the cones φ

= π/4 and φ = 3π/4
Mathematics
1 answer:
MrRissso [65]3 years ago
4 0

Answer:

So,  the volume is:

\boxed{V=\frac{128\sqrt{2}\pi}{3}}

Step-by-step explanation:

We get the limits of integration:

R=\left\lbrace(\rho, \varphi, \theta):\, 0\leq \rho \leq  4,\, \frac{\pi}{4}\leq \varphi\leq \frac{3\pi}{4},\, 0\leq \theta \leq 2\pi\right\rbrace

We use the spherical coordinates and  we calculate a triple integral:

V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\int_0^4  \rho^2 \sin \varphi \, d\rho\, d\varphi\, d\theta\\\\V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sin \varphi \left[\frac{\rho^3}{3}\right]_0^4\, d\varphi\, d\theta\\\\V=\int_0^{2\pi}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \sin \varphi \cdot \frac{64}{3} \, d\varphi\, d\theta\\\\V=\frac{64}{3} \int_0^{2\pi} [-\cos \varphi]_{\frac{\pi}{4}}^{\frac{3\pi}{4}}  \, d\theta\\\\V=\frac{64}{3} \int_0^{2\pi} \sqrt{2} \, d\theta\\\\

we get:

V=\frac{64}{3} \int_0^{2\pi} \sqrt{2} \, d\theta\\\\V=\frac{64\sqrt{2}}{3}\cdot[\theta]_0^{2\pi}\\\\V=\frac{128\sqrt{2}\pi}{3}

So,  the volume is:

\boxed{V=\frac{128\sqrt{2}\pi}{3}}

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saw5 [17]

Answer:

The area is 452cm

Step-by-step explanation:

For the big square we have to multiply 15*18 and that is the area so-

15*18=270

Next we multiply 13 and 14

13*14=182

And now we add 270 and 182 to get the total area

270+182=452

7 0
3 years ago
Read 2 more answers
Data Set #1
Harman [31]

Answer:

The data table is attached below.

Step-by-step explanation:

The average of a set of data is the value that is a representative of the entire data set.

The formula to compute averages is:

\bar x=\frac{1}{n}\sum\limits^{n}_{i=1}{x_{i}}

Compute the average for drop 1 as follows:

\bar x_{1}=\frac{1}{3}\times[10+11+9]=10

Compute the average for drop 2 as follows:

\bar x_{2}=\frac{1}{3}\times[29+31+30]=30

Compute the average for drop 3 as follows:

\bar x_{3}=\frac{1}{3}\times[59+58+61]=59.33

Compute the average for drop 4 as follows:

\bar x_{4}=\frac{1}{3}\times[102+100+98]=100

Compute the average for drop 5 as follows:

\bar x_{5}=\frac{1}{3}\times[122+125+127]=124.67

The data table is attached below.

3 0
3 years ago
A farmer has 200m of fencing. What is the area of the largest square paddock that could be fenced
zheka24 [161]
As perimeter of a square is 4*x, it means x=Perimeter/4=200m/4=50m.Now area of the square=x^2=50m^2=2500sq. meter.
6 0
3 years ago
Show how u got the answer 309×42
luda_lava [24]
                3            
                1
                390
              x  42
               -------
                1282
              1564
          ----------------
                       
the answer would be 16380
Hope this helped :)
4 0
3 years ago
The sand falls at a rate of 60 cubic millimeters per second. How much time do you have to answer the question
e-lub [12.9K]
You Have -3 Seconds FAM 
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