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MissTica
3 years ago
10

Transform the following polar equation into an equation in rectangular coordinates: r=2 cos theta A. x + y = 2 B. y = 2x C. x =

2 D. (x-1)^2+y^2=1
Mathematics
2 answers:
Mila [183]3 years ago
8 0
r=2\cos\theta
r^2=2r\cos\theta
\implies x^2+y^2=2x
x^2-2x+y^2=0
x^2-2x+1+y^2=1
(x-1)^2+y^2=1
yuradex [85]3 years ago
8 0

Answer:

The correct option is D. (x - 1)² + y² = 1

Step-by-step explanation:

Given the equation in polar coordinates : r = 2 cosθ

To change the given equation in rectangular coordinates, we use the relation : x = r cosθ and y = r sinθ

⇒ x² + y² = r²

Now, r = 2 cosθ

Multiplying by r on both the sides

⇒ r² = 2r cosθ

⇒ x² + y² = 2x

⇒ x² - 2x + y² = 0

Making the variable x, the complete square by adding 1 on both the sides

⇒ x² - 2x + 1 + y² = 0 + 1

⇒ (x - 1)² + y² = 1

Hence, This is our required rectangular coordinate form of the given equation.

Therefore, The correct option is D. (x - 1)² + y² = 1

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<h2>Answer:</h2>

<u>First Part</u>

Given that

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We have that

Volume = \frac{4}{3} \pi r^{3} =  \frac{4}{3} \pi (\frac{Diameter}{2})^{3} =  \frac{4}{3} \pi 9^{3} = 972\pi cm^{3} \approx 3053.63 cm^{3}

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If the Diameter were reduced by half we have that

Volume = \frac{4}{3} \pi r^{3} =  \frac{4}{3} \pi (\frac{r}{2}) ^{3} = \frac{\frac{4}{3} \pi r^{3}}{8}

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<h2>Step-by-step explanation:</h2>

<u>First Part</u>

Gather Information

Diameter = 18cm

Volume = \frac{4}{3} \pi r^{3}

Calculate Radius from Diameter

Radius = \frac{Diameter}{2} = \frac{18}{2} = 9

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Volume = \frac{4}{3} \pi r^{3} =  \frac{4}{3} \pi 9^{3}

Before starting any calculation, we try to simplify everything we can by expanding the exponent and then factoring one of the 9s

Volume = \frac{4}{3} \pi 9^{3} = \frac{4}{3} \pi 9 * 9 * 9 = \frac{4}{3} \pi 9 * 9 * 3 * 3

We can see now that one of the 3s can be already divided by the 3 in the denominator

Volume = \frac{4}{3} \pi 9 * 9 * 3 * 3 = 4 \pi 9 * 9 * 3

Finally, since we can't simplify anymore we just calculate it's volume

Volume = 4 \pi 9 * 9 * 3 = 12 \pi * 9 * 9 = 12 * 81 \pi = 972 \pi cm^{3}

Volume \approx 3053.63 cm^{3}

<u>Second Part</u>

Understanding how the Diameter reduced by half would change the Radius

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Understanding how the Radius now changes the Volume

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But we can see that the numerator is exactly the original Volume!

This shows us that the Volume would be  \frac{1}{8} of the original Volume if the Diameter were reduced by half.

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