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Liono4ka [1.6K]
3 years ago
12

Use spherical coordinates. evaluate (9 − x2 − y2) dv, where h is the solid hemisphere x2 + y2 + z2 ≤ 4, z ≥ 0.

Mathematics
1 answer:
avanturin [10]3 years ago
4 0
In spherical coordinates, we have

\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}

which gives volume element

\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta

and so the triple integral is given by

\displaystyle\iiint_H(9-x^2-y^2)\,\mathrm dV
=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=0}^{\varphi=\pi/2}\int_{\rho=0}^{\rho=2}(9-\rho^2\sin^2\varphi)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta
=\displaystyle2\pi\int_{\varphi=0}^{\varphi=\pi/2}\int_{\rho=0}^{\rho=2}(9\rho^2\sin\varphi-\rho^4\sin^3\varphi)\,\mathrm d\rho\,\mathrm d\varphi
=\dfrac{592\pi}{15}
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The volume of a right cone is 3525\piπ units^3
Greeley [361]

The height of the right cone given its volume and its diameter is  47 units.

<h3>What is the height?</h3>

A cone is a three-dimensional object that has a circular base and a vertex.

Height of the cone = Volume of a cone ÷ 1/3(πr²)

Where:

  • π = 22/7
  • r = radius = diameter / 2 = 30 / 2 = 15 units
  • h = height

3525π÷ (1/3 x π x 15²) = 47 units

To learn more about the volume of a cone, please check: brainly.com/question/13705125

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8 0
2 years ago
HELP URGENT I WILL GIVE 20 POINTS
Lunna [17]

h = 9.2 cm

Solution:

Base of smaller triangle = 5 cm

Height of smaller triangle = 2.3 cm

Base of larger triangle = 15 cm + 5 cm = 20 cm

Height of larger triangle = h

$\frac{2.3}{5}=\frac{h}{15+5}

$\frac{2.3}{5}=\frac{h}{20}

Multiply by 20 on both sides.

$20 \times \frac{2.3}{5}=\frac{h}{20} \times 20

$\frac{46}{5}=h

9.2 = h

So that Ryan should have add 15 and 5 to get a denominator of 20 on the right sides of the first line. Doing so gives the correct value, h = 9.2 cm.

8 0
3 years ago
What is A decreased by 20% and then increased by 20% of that? Please show your work.
Eduardwww [97]

Answer:

A - (20% of A) = 80% of A + (20% of (80% of A) = 96% of A

So if A is 100:

100 - (.2 * 100) = 80 + (.2 * 80) = 96

Step-by-step explanation:

3 0
3 years ago
Help please Find the surface area, including the floor, of his tent.
natita [175]

Answer:

52.8 m. sq.

Step-by-step explanation:

2.6 x 1.5 + 1.5 (3) = 7.8 / 2 = 3.9 x 2 = 7.8 (Triangles)

3 x 5 = 15 x 2 = 30 (Side Rectangles)

5 x 1.5 + 1.5 (3) = 15

15 + 30 + 7.8 = 52.8 m. sq.

Hope this helps!

If something is wrong, please let me know.

3 0
3 years ago
The range of three numbers is 17.the mode is 24 and the median is 24 ? What are the three numbers?
vivado [14]

The answer is 7, 24, 24.

The median is the middle number.

The mode is the number which occurs the most.

The range is the difference between the largest number and the smallest number.

Since 24 is the middle number, it is the median.

Since 24 is also the number which occurs the most often, it is also the mode.

To find range, do 24-7 which is equal to seventeen.

<em>Hope it helps</em>


3 0
3 years ago
Read 2 more answers
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