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3 years ago

A rectangle has an area of 16in^2 and a perimeter of 20 in. Find the dimension of the rectangle.

Mathematics

1 answer:

Anni [7]3 years ago

3 0

Answer:

8 in x 2 in

Step-by-step explanation:

8 x 2 = 16

8 + 8 + 2 + 2 = 20

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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

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

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3 years ago

The taste test for PTC (phenylthiourea) is a common class demonstration in the study of genetics. It is known that 70% of the Am

AysviL [449]

Answer:

The variance for the number of tasters is 4.2

Step-by-step explanation:

For each person, there are only two possible outcomes. Either they are tasters, or they are not. The probability of a person being a taster is independent of any other person. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The variance of the binomial distribution is:

$V(X) = np(1-p)$

It is known that 70% of the American people are "tasters" with the rest are "non-tasters". Suppose a genetics class of size 20

This means that $p = 0.7, n = 20$

$V(X) = np(1-p) = 20*0.7*0.3 = 4.2$

The variance for the number of tasters is 4.2

7 0

3 years ago

Look at the diagram. Which of the following is another name 22? с. А D 1/2 B

stiks02 [169]

Answer:

i think another name for angel 2 is CBD or DBC i ain't sure

Step-by-step explanation:

6 0

3 years ago

Find real numbers a, b, and c so that the graph of the function y equals ax squared plus bx plus c contains the points left par

andrezito [222]

Answer:

ax 2+bx+c,where a≠0

Step-by-step explanation:

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3 years ago

4/3 + -1/6 + 13/12. Please answer step by step if possible. Thanks.

strojnjashka [21]

Answer:

9/4

Step-by-step explanation:

We follow bodmas

4/3 +( -1/6 + 13/12)

( lcm = 12)

( -2+ 13/12)

( 11/ 12)

4/3 + ( 11/12)

4/3 + 11/12

lcm = 12 also

and that will equal to

=16 + 11/ 12

= 27/ 12

divide by 3 to simplest form

= 9/4

6 0

2 years ago

Read 2 more answers