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19 Sep 2017, 11:29
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D
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A pizza with diameter of 12 inches is split into 8 equally sized pieces. (There is a figure here I cannot insert for some reason: imagine a circle with 8 points on the perimeter: A through H. The center is point "O"). 4 non-adjacent pieces are removed. What is the perimeter AOBCODEOFGOHA of the pizza now, including the inside edges of the slices?
(A) \(48\pi+48\)
(B) \(24\pi +48\)
(C) \(24\pi +24\)
(D) \(6\pi +48\)
(E) \(6\pi +24\)
Attachment:
2023-01-04_16-39-19.png [ 21.15 KiB | Viewed 2727 times ]
19 Sep 2017, 11:46
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LakerFan24
Attachment:
pizza.png [ 16.65 KiB | Viewed 13423 times ]
Gray areas are the remaining slices; we need their perimeter, totaled.
The pizza has had every other slice removed. Each slice has two straight edges and an arc as its perimeter.
I. Perimeter = All remaining arc lengths + all straight edges
Because the four remaining slices' perimeters are equal, find the perimeter of all four by finding the sum of the lengths four remaining arcs, and the sum of the lengths of the eight remaining straight sides.
1) Find the total length of the arcs that are left
The circumference of the pizza is \(\pi*d\), \(d = 12\), so circumference = \(12\pi\)
Half of the circumference has been removed, so the total length for the four arcs remaining is \(6\pi\)
2) Find the total of the straight edges' lengths, all of which are radii.
Radius = \(\frac{d}{2}= 6\)
There are 8 edges. If you've sketched, count. If not, each of four pieces has two straight edges, and 4 * 2 straight edges = 8 straight edges
\(8*6 = 48\) is the total of lengths of the straight edges
3) Add both totals. The perimeter is the total of the four arcs, and the total of the eight straight edges:
\(6\pi + 48\)
II. Perimeter needed = Perimeter of each remaining piece * 4
1) Find arc length of one remaining slice
The circumference of the pizza is \(\pi*d\), \(d = 12\), so circumference = \(12\pi\)
Each piece is \(\frac{1}{8}\) of the pizza, so its arc length is \(\frac{1}{8}\) of the circumference
Each piece's arc length = \(12\pi\) * \(\frac{1}{8}\) = \(\frac{12\pi}{8}\) = \(\frac{3\pi}{2}\)
2) Find the other two parts of a piece's perimeter: two straight edges, both of which are radii
If \(d = 12, r = 6\). Each piece has two straight edges. Total for the two sides: \(6 + 6 = 12\)
3) Each piece's perimeter is \(\frac{3\pi}{2}\)\(+ 12\)
4) There are four pieces, multiply their perimeter by 4
(4) (\(\frac{3\pi}{2} + 12\)) =
\(6\pi + 48\)
ANSWER D
19 Sep 2017, 14:45
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Given : Diameter of pizza = 12 inch => radius = 6 inch
Pizza cut into 8 pieces.
4 non-adjacent are removed.
Attached figure shows pizza layout with X marked for removed pieces.
Find: perimeter of remaining portion ?
So basically perimeter of 4 pizza pieces attached at center
Perimeter of 1 piece = 2 * side + Circular portion
Perimeter of 4 pieces = 4*( 2 * side + Circular portion)
1 side = radius of circle = 6 inches
Circular portion = 1/8 th of whole circumference = \(\frac{1}{8} * (2*pi*r)\) = \(\frac{1}{8} * (2*pi*6)\)
So perimeter of remaining 4 pieces combined = 4*( 2 * 6 + \(\frac{1}{8} * (2*pi*6)\) ) = 48 + 6*pi
[ Here insted of finding 1/8th of circumference , we can also say as 50% portion is removed , remaining circular portion (that is of all 4 pieces ) is 1/2 of total circumference ]
Answer: D
Pizza cut into 8 pieces.
4 non-adjacent are removed.
Attached figure shows pizza layout with X marked for removed pieces.
Find: perimeter of remaining portion ?
So basically perimeter of 4 pizza pieces attached at center
Perimeter of 1 piece = 2 * side + Circular portion
Perimeter of 4 pieces = 4*( 2 * side + Circular portion)
1 side = radius of circle = 6 inches
Circular portion = 1/8 th of whole circumference = \(\frac{1}{8} * (2*pi*r)\) = \(\frac{1}{8} * (2*pi*6)\)
So perimeter of remaining 4 pieces combined = 4*( 2 * 6 + \(\frac{1}{8} * (2*pi*6)\) ) = 48 + 6*pi
[ Here insted of finding 1/8th of circumference , we can also say as 50% portion is removed , remaining circular portion (that is of all 4 pieces ) is 1/2 of total circumference ]
Answer: D
Attachments
Pizza layout.png [ 8.4 KiB | Viewed 12739 times ]
