How to find the percentage of a sector from an angle - GRE Quantitative Reasoning

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Question

A given pizza with a -inch diameter has calories. A baker cuts the pizza using a $25^{\circ}$ angle for each piece. If Susan eats five such pieces, how many calories does she consume? Round to the nearest calorie.

Answer

To solve this, notice that one piece of pizza comprises $\frac{25}{360}$ of the total pie or (reducing) $\frac{5}{72}$ of a pie. Now, if Susan buys five slices, she gets:

$\frac{5}{72} \cdot 5 = \frac{25}{72}$ of the pie. If the complete pie contains calories, she will then eat the following amount of calories:

$\frac{25}{72} \cdot 1450 = 503.47$

Rounding, this is calories.

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Question

A circle of radius , for an angle , has an arc length of $3\pi$ . What is the angle ?

Answer

The formula for an arc length, , of a circle of a given radius, , and a given angle, is:

$l=2\pi r(\frac{x}{360})$

Note that $2\pi r$ is the circumference of the circle.

Conversely, for a known arc length and unknown angle, this equation can be rewritten as follows:

$x=360(\frac{l}{2\pi r})$

Plugging in the given values, it is therefore possible to find the missing angle:

$x=360(\frac{3\pi }{2\pi(5 )})=108^{\circ}$

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Question

Find the radius of the given circle:

Answer

To solve this problem, realize that an inscribed angle (an angle formed by two chords) is equal to twice the central angle formed by connecting the origin to the inscribed angle's endpoints:

Now, the formula for an arc length, , of a circle of a given radius, , and a given angle, is:

$l=2\pi r(\frac{x}{360})$

Since the radius is the unknown, this equation can be rewritten as:

$r=(\frac{360}{x})(\frac{l}{2\pi })$

Plugging in our values, we find:

$r=\frac{360}{60}\frac{6\pi}{2\pi}=18$

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