How to find the length of an arc - GRE Quantitative Reasoning
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What is the perimeter of a pie piece if the pie is sliced into 40 degree pieces and its area is 361π?
What is the perimeter of a pie piece if the pie is sliced into 40 degree pieces and its area is 361π?
The perimeter of a given pie-piece will be equal to 2 radii plus the outer arc (which is a percentage of the circumference). For our piece, this arc will be 40/360 or 1/9 of the circumference. If the area is 361π, this means πr2 = 361 and that the radius is 19. The total circumference is therefore 2 * 19 * π or 38π.
The total perimeter of the pie piece is therefore 2 * r + (1 / 9) * c = 2 * 19 + (1/9) * 38π = 38 + 38π/9
The perimeter of a given pie-piece will be equal to 2 radii plus the outer arc (which is a percentage of the circumference). For our piece, this arc will be 40/360 or 1/9 of the circumference. If the area is 361π, this means πr2 = 361 and that the radius is 19. The total circumference is therefore 2 * 19 * π or 38π.
The total perimeter of the pie piece is therefore 2 * r + (1 / 9) * c = 2 * 19 + (1/9) * 38π = 38 + 38π/9
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What is the length of the arc of a circle with radius 10 that traces a 50 degree angle?
What is the length of the arc of a circle with radius 10 that traces a 50 degree angle?
length of an arc = (degrees * 2_πr)/360 = (50 * 2_π * 10)/360 = 25_π_/9
length of an arc = (degrees * 2_πr)/360 = (50 * 2_π * 10)/360 = 25_π_/9
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An ant walks around the edge of circular pizzas left on the counter of a pizza shop. On most days, it is shaken off the pizza before it manages to walk the complete distance.
Quantity A: The distance covered by the ant when walking over four slices of a pizza with a diameter of 
and 
equally-sized pieces.
Quantity B: The distance covered by the ant when walking over a complete personal pizza with a diameter of 
inches.
What can we say about the two quantities?
An ant walks around the edge of circular pizzas left on the counter of a pizza shop. On most days, it is shaken off the pizza before it manages to walk the complete distance.
Quantity A: The distance covered by the ant when walking over four slices of a pizza with a diameter of
Quantity B: The distance covered by the ant when walking over a complete personal pizza with a diameter of
What can we say about the two quantities?
Let's compute each quantity.
Quantity A
This is a little bit more difficult than quantity B. It requires us to compute an arc length, for the ant does not walk around the whole pizza; however, this is not very hard. What we know is that the ant walks around 
, or 
, of the pizza.
Now, we know the circumference of a circle is 
or 
For our example, let's use the latter. Since 
, we know:
However, our ant walks around only part of this, namely:
Quantity B
This is really just a matter of computing the circumference of the circle. For our value, we know this to be:
Now, we can compare these by taking quantity A and reducing the fraction to be 
. Thus, we know that Quantity B is larger than Quantity A.
Let's compute each quantity.
Quantity A
This is a little bit more difficult than quantity B. It requires us to compute an arc length, for the ant does not walk around the whole pizza; however, this is not very hard. What we know is that the ant walks around
Now, we know the circumference of a circle is
For our example, let's use the latter. Since
However, our ant walks around only part of this, namely:
Quantity B
This is really just a matter of computing the circumference of the circle. For our value, we know this to be:
Now, we can compare these by taking quantity A and reducing the fraction to be
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