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2 months ago

HELP!!! Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 4 cos 3θ

Mathematics

2 answers:

Zina [12.3K]2 months ago

6 0

Response:

The graph exhibits symmetry concerning the x-axis.

Step-by-step outline:

Symmetry about the polar axis (x-axis) is observed when replacing (r,θ) with (r,-θ), resulting in an equation identical to the original.

For the y-axis, replacing (r,θ) with (-r,-θ) gives the same equation as the original.

For symmetry around the pole (origin), a switch of (r,θ) to (-r,θ) must yield the same original equation.

We have the following equation:

$r=4\cos 3\theta$

When substituting (r,θ) with (r,-θ), the result is:

$r=4\cos 3(-\theta)\\\\i.e.\\\\r=4\cos (-3\theta)\\\\i.e.\\\\r=4\cos 3\theta$

( Since, we are aware that:

$\cos (-\tehta)=\cos \theta$ )

Therefore, the graph is symmetric with respect to the x-axis.

For when we substitute (r,θ) with (-r,-θ)

$-r=4\cos 3(-\theta)\\\\i.e.\\\\-r=4\cos (-3\theta)\\\\i.e.\\\\-r=4\cos 3\theta\\\\i.e.\\\\r=-4\cos 3\theta$

In this case, we see that the graphs are not equivalent upon performing this substitution.

Consequently, the graph lacks symmetry concerning the y-axis.

Upon substituting (r,θ) with (-r,θ), we find:

$-r=4\cos 3\theta\\\\i.e.\\\\r=-4\cos 3\theta$

Again, it's clear that the resulting graphs do not equate.

Thus, the graph does not exhibit symmetry with respect to the pole (origin).[TAG_63]]

Leona [12.6K]2 months ago

3 0

To evaluate y-symmetry, let’s define r = 5 cos 3θ and set θ=π−θ which implies that r=5cos3θθ=(π−θ)r=5cos(3(π−θ))⟹r=5cos(3π−3θ)

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1 month ago

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Add 0.0025 kg, 1750 mg, 2.25 g, and 825,000 μg, and express the answer in grams.

lawyer [12517]

Answer:

6.75g

Step-by-step explanation:

The initial step is to convert all values to grams using rules of three, followed by summation.

In a rule of three calculation, the first step is identifying the measures and how they interrelate; whether their connection is direct or inverse.

In direct relationships, as one measure increases, so does the other.

In inverse relationships, as one measure increases, the other decreases.

Problems involving unit conversion, like this, illustrate a direct relationship between measures.

First step: convert 0.0025kg to grams

One kilogram equals 1000 grams, so:

1kg - 1000g

0.0025kg - xg

x = 1000*0.0025 = 2.5g

Thus, 0.0025kg is equal to 2.5g

Second step: handle 1750 mg

One gram consists of 1000mg, so:

1g - 1000mg

xg - 1750mg

1000x = 1750

$x = \frac{1750}{1000}$

x = 1.75g

Therefore, 1750 mg converts to 1.75g

2.25g is fine

Third step: convert 825,000 μg to grams

One gram consists of 1,000,000 micrograms, so:

1g - 1,000,000 μg

xg - 825,000 μg

1,000,000x = 825,000

$x = \frac{825,000}{1,000,000}$

x = 0.25g

So, 825,000 μg equals 0.25g

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3 months ago

A box has 14 camera of which 6 are refurbished and 8 are new. If four of these 14 cameras are selected at random without replace

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Answer:

160/1001, 175/1001

Step-by-step explanation:

i) We calculate:

₈C₁ methods to select 1 new camera from a selection of 8

₆C₃ methods to select 3 refurbished cameras from a selection of 8

₁₄C₄ methods to select 4 cameras from the total of 14 cameras

The probability formula is:

P = ₈C₁ ₆C₃ / ₁₄C₄

P = 8×20 / 1001

P = 160 / 1001

P ≈ 0.160

ii) For at most one new camera, it means we want either one new camera or none at all. We've calculated the probability of selecting one new camera already. The probability of not selecting any new camera is equivalent to selecting 4 refurbished cameras:

P = ₆C₄ / ₁₄C₄

P = 15 / 1001

Therefore, the combined probability is:

P = 160/1001 + 15/1001

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P ≈ 0.175

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3 months ago

The line width of a tool used for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micromete

zzz [12365]

Answer:

a. 0.82%

b. 71.11%

c. 0.564 micrometer

Step-by-step explanation:

To find the z value for each measurement, we must calculate and determine the percentage they correspond to, and the difference will give us the percentage between those two statistics.

The equation for z is:

z = (x - m) / (sd)

Here, x is the value being evaluated, m denotes the mean, and sd represents the standard deviation.

For 0.62 copies, we calculate:

z = (0.62 - 0.5) / (0.05)

z = 2.4

This translates to 0.9918.

Therefore, p (x > 0.62) = 1 - 0.9918

p (x > 0.62) = 0.0082 = 0.82%

For 0.47 copies, we have:

z = (0.47 - 0.5) / (0.05)

z = -0.6, which equates to 0.2742.

For 0.63 copies:

z = (0.63 - 0.5) / (0.05)

z = -2.6, which yields 0.9953.

Hence, p (0.47 > x > 0.63) = 0.9953 - 0.2742

p (0.47 > x > 0.63) = 0.7211 = 72.11 %

For x copies, we find:

p = 0.9 corresponds to z = 1.28.

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From which we derive:

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3 months ago

Two thirds of a straight angle

PIT_PIT [12445]

Answer:

Two thirds of a straight angle is 120°

Explanation:

First, we define a straight angle:

A straight angle measures 180°

Now, we need to find two thirds of that straight angle

To compute the fraction, multiply the straight angle's measurement by $\frac{2}{3}$

Thus:

$\frac{2}{3}$ of a straight angle = $\frac{2}{3} * 180 = 120$ degrees

Hope this helps:)

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1 month ago