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2 years ago

Find the volume of the cone

Mathematics

1 answer:

Anuta_ua [19.1K]2 years ago

8 0

Volume= 37. 7

V=π r² h/3 = π · 22 · 9/3 ≈37.69911

I hope this helps!

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Sketch the domain D bounded by y = x^2, y = (1/2)x^2, and y=6x. Use a change of variables with the map x = uv, y = u^2 (for u ?

cluponka [151]

Under the given transformation, the Jacobian and its determinant are

$\begin{cases}x=uv\\y=u^2\end{cases}\implies J=\begin{bmatrix}v&u\\2u&0\end{bmatrix}\implies|\det J|=2u^2$

so that

$\displaystyle\iint_D\frac{\mathrm dx\,\mathrm dy}y=\iint_{D'}\frac{2u^2}{u^2}\,\mathrm du\,\mathrm dv=2\iint_{D'}\mathrm du\,\mathrm dv$

where $D'$ is the region $D$ transformed into the $u$ - $v$ plane. The remaining integral is the twice the area of $D'$ .

Now, the integral over $D$ is

$\displaystyle\iint_D\frac{\mathrm dx\,\mathrm dy}y=\left\{\int_0^6\int_{x^2/2}^{x^2}+\int_6^{12}\int_{x^2/2}^{6x}\right\}\frac{\mathrm dx\,\mathrm dy}y$

but through the given transformation, the boundary of $D'$ is the set of equations,

$\begin{array}{l}y=x^2\implies u^2=u^2v^2\implies v^2=1\implies v=\pm1\\y=\frac{x^2}2\implies u^2=\frac{u^2v^2}2\implies v^2=2\implies v=\pm\sqrt2\\y=6x\implies u^2=6uv\implies u=6v\end{array}$

We require that $u>0$ , and the last equation tells us that we would also need $v>0$ . This means $1\le v\le\sqrt2$ and $0$ , so that the integral over $D'$ is

$\displaystyle2\iint_{D'}\mathrm du\,\mathrm dv=2\int_1^{\sqrt2}\int_0^{6v}\mathrm du\,\mathrm dv=\boxed6$

4 0

2 years ago

Which word phrases represent the variable expression 7d? Choose all answers that are correct. A. the product of 7 and a number B

amm1812

A and D are correct.

8 0

3 years ago

Read 2 more answers

How many feet long is the shadow of a 15-foot flagpole if a 5-foot woman standing at the base of the flagpole has a shadow that

Vikentia [17]

The height of flagpole is 12.5 feet.

Step-by-step explanation:

Given,

Height of woman = 5 foot

Shadow of woman = 6 feet

Ratio of height to shadow = 5:6

Shadow of flagpole = 15 foot

Height of flagpole = x

Ratio of height to shadow = x:15

Using proportion;

Ratio of height to shadow :: Ratio of height to shadow

$5:6::x:15\\$

Product of mean = Product of extreme

$6*x=15*5\\6x=75$

Dividing both sides by 6

$\frac{6x}{6}=\frac{75}{6}\\x=12.5$

The height of flagpole is 12.5 feet.

Keywords: ratio, proportion

Learn more about proportion at:

#LearnwithBrainly

8 0

2 years ago

Write the given roots to quadratic equation

Svet_ta [14]

Answer:

α + β = - ba and αβ = ca.

⇒ x2 + bax + ca = 0 (Since, a ≠ 0)

⇒ x2 - (α + β)x + αβ = 0, [Since, α + β = -ba and αβ = ca]

Step-by-step explanation:

pls mark brainlist

4 0

2 years ago

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Help!! Thank you so much!!

Orlov [11]

The volume we're looking for is the volume of both cones in the figure.

The volume of a cone is $V=\pi r^2\frac{h}{3}$ .

So, $V_{tot} = V_{1} + V_2$ .

<u>Cone 1's variables:</u>

r = 2.6

h = 5

<u>Cone 2's variables:</u>

r = 2.6

h = 3

Now we can just plug and chug!

$V_{tot} = [(3.14)(2.6^2)(\frac{5}{3} )] + [(3.14)(2.6^2)\frac{3}{3} )]\\V_{tot} = [(3.14)(6.76)(1.67)] + [(3.14)(6.76)(1)]\\V_{tot} = (35.45) + (21.23)\\V_{tot} = 56.68$

V = c. 56.6 cubic units

5 0

2 years ago