Find the solution to the system of equations. y = 2x + 2 ([?], [?])

Use cylindrical coordinates. find the volume of the solid that is enclosed by the cone z = x2 + y2 and the sphere x2 + y2 + z2 =

sashaice [31]

Let $R$ be the solid. Then the volume is

$\displaystyle\iiint_R\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$

which follows from the facts that

$\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$
(by computing the Jacobian)

and

$z=x^2+y^2=r^2\implies z+z^2=72\implies z=-9\text{ or }z=8$
(we take the positive solution, since it's clear that $R$ lies above the $x$ - $y$ plane)
$r^2+z^2=72\implies z=\pm\sqrt{72-r^2}$
(again, taking the positive root for the same reason)
$z=r^2\implies 8=r^2\implies r=\sqrt8$

$\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}r(\sqrt{72-r^2}-r^2)\,\mathrm dr$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}(r\sqrt{72-r^2}-r^3)\,\mathrm dr$
$=\dfrac{32(27\sqrt2-35)\pi}3$

7 0

3 years ago

Ravi has some money. If he buys 40 doughnuts, he will have $5 left. If he buys 50 doughnuts,he will need another $15. How much d

anastassius [24]

Difference in doughnuts = 50 - 40 = 10
Difference in money = 15 + 5 = 20

20 ÷ 10 = $2
Each doughnut costs $2

Check:
40 x 2 + 5 = $85
50 x 2 -15 = $85
Ravi has $85 at first.

Each doughnut costs $2

4 0

3 years ago

PLEASE HELP!!!! 20 POINTS!!!!

galben [10]

These are two questions and two answers.

Question 1) Which of the following polar equations is equivalent to the parametric equations below?

 x=t²
y=2t

Answer: option A.) r = 4cot(theta)csc(theta)


Explanation:

1) Polar coordinates ⇒ x = r cosθ and y = r sinθ

2) replace x and y in the parametric equations:

r cosθ = t²
r sinθ = 2t

3) work r sinθ = 2t

r sinθ/2 = t
(r sinθ / 2)² = t²

4) equal both expressions for t²

r cos θ = (r sin θ / 2 )²

5) simplify

r cos θ = r² (sin θ)² / 4

4 = r (sinθ)² / cos θ

r = 4 cosθ / (sinθ)²

r = 4 cot θ csc θ ↔ which is the option A.

Question 2) Which polar equation is equivalent to the parametric equations below?

 x=sin(theta)cos(theta)+cos(theta)
y=sin^2(theta)+sin(theta)

Answer: option B) r = sinθ + 1

Explanation:

1) Polar coordinates ⇒ x = r cosθ, and y = r sinθ

2) replace x and y in the parametric equations:

a) r cosθ = sin(θ)cos(θ)+cos(θ)
 b) r sinθ =sin²(θ)+sin(θ)

3) work both equations

a) r cosθ = sin(θ)cos(θ)+cos(θ) ⇒ r cosθ = cosθ [ sin θ + 1] ⇒ r = sinθ + 1

 b) r sinθ =sin²(θ)+sin(θ) ⇒ r sinθ = sinθ [sinθ + 1] ⇒ r = sinθ + 1


Therefore, the answer is r = sinθ + 1 which is the option B.

6 0

3 years ago

Read 2 more answers

10+3c≤1 ????????????????????????

stiks02 [169]

Answer:

c<-3

Step-by-step explanation:

move the constant to the right-hand side and change its sing

Then canculate the difference

Then divide both sides for the inequality of 3

6 0

3 years ago

The total area of Wisconsin is 65498 square miles. Of that about 80% is land area. About how many of Wisconsin is not land area

geniusboy [140]

Answer:

52398.4 square miles

Step-by-step explanation:

7 0

4 years ago

Read 2 more answers

Find the solution to the system of equations. y = 2x + 2 ([?], [?])​

Find the solution to the system of equations. y = 2x + 2 ([?], [?])