How is the frequency table with intervals similar to the frequency tables and How is it different?

Mathematics

1 answer:

stepan [7]3 years ago

4 0

I would help but I’m not good with graphs

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The graphs of the polar curves r = 4 and r = 3 + 2cosθ are shown in the figure above. The curves intersect at θ = π/3 and θ = 5π

Gennadij [26K]

(a)

$\displaystyle \frac{1}{2} \cdot \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(4^2 - (3 + 2\cos\theta)^2 \right) \, d\theta$

or, via symmetry

$\displaystyle\frac{1}{2} \cdot 2 \int_{\frac{\pi}{3}}^{\pi} \left(4^2 - (3 + 2\cos\theta)^2 \right) \, d\theta$

____________

(b)

By the chain rule:

$\displaystyle \frac{dy}{dx} = \frac{ dy/ d\theta}{ dx/ d\theta}$

For polar coordinates, x = rcosθ and y = rsinθ. Since
r = 3 + 2cosθ, it follows that

$x = (3 + 2\cos\theta) \cos \theta \\ 
y = (3 + 2\cos\theta) \sin \theta$

Differentiating with respect to theta

$\begin{aligned}
\displaystyle \frac{dy}{dx} &= \frac{ dy/ d\theta}{ dx/ d\theta} \\
&= \frac{(3 + 2\cos\theta)(\cos\theta) + (-2\sin\theta)(\sin\theta)}{(3 + 2\cos\theta)(-\sin\theta) + (-2\sin\theta)(\cos\theta)} \\ \\
\left.\frac{dy}{dx}\right_{\theta = \frac{\pi}{2}}
&= 2/3
\end{aligned}$

2/3 is the slope

____________

(c)

"distance between the particle and the origin increases at a constant rate of 3 units per second" implies dr/dt = 3

Angle θ and r are related via r = 3 + 2cosθ, so implicitly differentiating with respect to time

 $\displaystyle\frac{dr}{dt} = -2\sin\theta \frac{d\theta}{dt} \quad \stackrel{\theta = \pi/3}{\implies} \quad 3 = -2\left( \frac{\sqrt{3}}{2}}\right) \frac{d\theta}{dt} \implies \\ \\ \frac{d\theta}{dt} = -\sqrt{3} \text{ radians per second}$

5 0

3 years ago

What percent of 340 is 255?

ahrayia [7]

Answer:

255 is 75% of 340

Step-by-step explanation:

255 divided by 340 is .75, which can be transformed into a precentage as 75%

4 0

3 years ago

Help me pls and can u also explain if u want

Dafna1 [17]

Everything is "=" except 5/12.

Good luck! :)

8 0

3 years ago

Read 2 more answers

A study of dog owners in the city of Mayfield showed that 9% of the dog owners took their dog on vacation with them last year. M

Murrr4er [49]

9% of dog owners took their dog on vacation
Mayfield has approximately 4400 dog owners

Find 9% of 4400

0.09 * 4400 = 396

So 396 of Mayfield's dog owners took their dog on vacation with them last year.

3 0

3 years ago

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What 0.7 Divide would give you 4.97

valina [46]

Answer:

3.479

Step-by-step explanation:

4 0

4 years ago

How is the frequency table with intervals similar to the frequency tables and How is it different?​

How is the frequency table with intervals similar to the frequency tables and How is it different?