How to find the length of a radius - Math

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Question

The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?

(a) Twice the radius of Circle B

(b) The sum of the radius of Circle A and the radius of Circle C

Answer

Let be the radius of Circle A. Then its area is $\pi $r^{2}$$ .

The area of Circle B is $4\pi $r^{2}$ $=2^{2}$\pi $r^{2}$ = \pi $(2r)^{2}$$ , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or .

(a) Twice the radius of circle B is .

(b) The sum of the radii of Circles A and B is .

This makes (b) greater.

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Question

The time is now 1:45 PM. Since noon, the tip of the minute hand of a large clock has moved $\frac{14\pi}{3}$ feet. How long is the minute hand of the clock?

Answer

Every hour, the tip of the minute hand travels the circumference of a circle. Between noon and 1:45 PM, one and three-fourths hours pass, so the tip travels $1 $\frac{3}{4}$$ or $\frac{7}{4}$ times this circumference. The length of the minute hand is the radius of this circle , and the circumference of the circle is $C = 2 \pi r$ , so the distance the tip travels is $\frac{7}{4}$ this, or

$\frac{7}{4}$ C = $\frac{7}{4}$ \cdot 2 \pi r = $\frac{7 \pi r }{2}$

Set this equal to $\frac{14\pi}{3}$ feet:

$\frac{7 \pi r }{2}$ = $\frac{14\pi}{3}$

$\frac{7 \pi r }{2}$ \times $\frac{2}{7 \pi}$= $\frac{14\pi}{3}$\times $\frac{2}{7 \pi}$

$r= $\frac{2}{3}$\times $\frac{2}{1}$ = $\frac{4}{3}$ = 1 $\frac{1}{3}$$ feet.

This is equivalent to 1 foot 4 inches.

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Question

The tip of the minute hand of a giant clock has traveled $40 \pi$ feet since noon. It is now 2:30 PM. Which is the greater quantity?

(A) The length of the minute hand

(B) Three yards

Answer

Betwen noon and 2:30 PM, the minute hand has made two and one-half revolutions; that is, the tip of minute hand has traveled the circumference of its circle two and one-half times. Therefore,

$C \cdot 2 $\frac{1}{2}$ = 40 \pi$

$C = 40 \pi \div 2 $\frac{1}{2}$ = $\frac{40 \pi}{1}$ \div $\frac{5}{2}$ = $\frac{40 \pi}{1}$ \cdot $\frac{2}{5}$= $\frac{80 \pi}{5}$ = 16\pi$ feet.

The radius of this circle is the length of the minute hand. We can use the circumference formula to find this:

$2 \pi r = C$

$2 \pi r = 16 \pi$

$r = 16 \pi \div 2 \pi = 8$

The minute hand is eight feet long, which is less than three yards (nine feet), so (B0 is greater.

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Question

If the diameter of a circle is equal to , then what is the value of the radius?

Answer

Given that the radius is equal to half the diameter, the value of the radius would be equal to divided by 2. This gives us:

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Question

Compare the two quantities:

Quantity A: The radius of a circle with area of $64\pi$

Quantity B: The radius of a circle with circumference of $50\pi$

Answer

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * $r^2$$

$C=2\pir$

For our two quantities, we have:

Quantity A

$64\pi = \pi * $r^2$$

Therefore,

Taking the square root of both sides, we get:

Quantity B

$50\pi = 2 * \pi * r$

Therefore,

Therefore, quantity B is greater.

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Question

Compare the two quantities:

Quantity A: The radius of a circle with area of $144\pi$

Quantity B: The radius of a circle with circumference of $24\pi$

Answer

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * $r^2$$

$C=2\pir$

For our two quantities, we have:

Quantity A

$144\pi = \pi * $r^2$$

Therefore,

Taking the square root of both sides, we get:

Quantity B

$24\pi = 2 * \pi * r$

Therefore,

Therefore, the two quantities are equal.

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Question

The areas of five circles form an arithmetic sequence. The smallest circle has radius 4; the second smallest circle has radius 8. Give the radius of the largest circle.

Answer

The area of a circle with radius is $A = \pi $r^{2}$$ . Therefore, the areas of the circles with radii 4 and 8, respectively, are

and

The areas form an arithmetic sequence, the common difference of which is

$64 \pi - 16 \pi = 48 \pi$ .

The circles will have areas:

$16 \pi, 64 \pi, 112 \pi, 160 \pi , 208 \pi$

Since the area of the largest circle is $208 \pi$ , we can find the radius as follows:

The radius can be calculated now:

$\pi $r^{2}$ = 208 \pi$

$r = $\sqrt{208}$ = $\sqrt{16}$ \cdot $\sqrt{13}$ = 4 $\sqrt{13}$$

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Question

The area of a circle is . Give its radius in terms of .

(Assume is positive.)

Answer

The relation between the area of a circle and its radius is given by the formula

$A = \pi $r^{2}$$

Since

$A= $81x^{2}$$ :

We solve for :

$\pi $r^{2}$ = $81x^{2}$$

$$\frac{\pi $r^{2}$$ }{\pi}= $$\frac{81x^{2}$$}{\pi}$

Since is positive, as is :

$r =\frac{ \sqrt{81}$ \cdot $$\sqrt{x^{2}$$}}{$\sqrt{\pi}$} = $\frac{9x}{\sqrt{\pi}$}$

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Question

Find the length of the radius of a circle given the circumference is $6\pi$ .

Answer

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$$

$r=\frac{6\pi}{2\pi}$=3$

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Question

Find the radius of a circle given the circumference is $8\pi$ .

Answer

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$$

$r=\frac{8\pi}{2\pi}$=4$

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Question

Find the length of the radius of a circle given the diameter is 3.

Answer

To solve, simply use the formula for the diameter and solve for r. Thus,

Therefore, when solved for r,

$r=\frac{d}{2}$$

Plug in d and:

$r=\frac{3}{2}$$

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Question

The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate radius of the basketball?

Answer

To Find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. (The information given of 22 ounces is useless)

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Question

A circle with center (8, –5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?

Answer

For the circle to be tangent to the y-axis, it must have its outer edge on the axis. The center is 8 units from the edge.

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Question

A circle has an area of $49\pi \ $in^{2}$$ . What is the radius of the circle, in inches?

Answer

We know that the formula for the area of a circle is πr_2. Therefore, we must set 49_π equal to this formula to solve for the radius of the circle.

49_π_ = _πr_2

49 = _r_2

7 = r

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Question

Find the length of the radius given the circumference is 16

Answer

To solve, simply use the formula of the circumference and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$$

$r=\frac{16}{2\pi}$=\frac{8}{\pi}$$

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Question

A circle has an area of 36π inches. What is the radius of the circle, in inches?

Answer

We know that the formula for the area of a circle is π_r_2. Therefore, we must set 36π equal to this formula to solve for the radius of the circle.

36π = π_r_2

36 = _r_2

6 = r

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Question

Circle X is divided into 3 sections: A, B, and C. The 3 sections are equal in area. If the area of section C is 12π, what is the radius of the circle?

Circle X

Answer

Find the total area of the circle, then use the area formula to find the radius.

Area of section A = section B = section C

Area of circle X = A + B + C = 12π+ 12π + 12π = 36π

Area of circle = where r is the radius of the circle

36π = πr2

36 = r2

√36 = r

6 = r

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Question

The area of a circle is one square yard. Give its radius in inches, to the nearest tenth of an inch.

Answer

The area of a circle is

$A = \pi $r^{2}$$

Substitute 1 for :

$1 = \pi $r^{2}$$

$r = $\sqrt{\frac{1}{\pi }$}$

This is the radius in yards. The radius in inches is 36 times this.

$36$\sqrt{\frac{1}{\pi }$} \approx 20.3$

20.3 inches is the radius.

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Question

The circle shown below has an area equal to $49\pi$ . What is the length of the radius, , of this circle?

Answer

The formula for the area of a circle is $A=\pi $R^{2}$$ . We can fill in what we know, the area, and then solve for the radius, .

$49\pi =\pi $R^{2}$$

Divide each side of the equation by $\pi$ :

Take the square root of each side:

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Question

A circle has a circumference of $32 \pi$ inches. What is the radius of the circle?

Answer

The circumference of a circle is given by $c = 2\pi * r$ , where is the circumference and is the radius.

Plug in the given circumference for and solve for :

$32 \pi = 2\pi * r$

$\frac{(32 \pi)}{2\pi}$ = $\frac{(2\pi * r)}{2\pi}$

$r= 16\ in$

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