Competition problem — Geometry — Maths Olympiad Prep

Library / /192 of 520

Geometry Difficulty 6.7 National olympiad Find the answer

An $8$ -foot by $10$ -foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Figure (Asymptote source)

unitsize(2cm);
defaultpen(linewidth(.8pt));

fill(unitsquare,gray);
filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);
filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);
filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);
filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);

\text{(A) 80 20 (B) 60 10 (C) 80 10 (D) 60 10 (E) 80 10}

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

1. Calculate the area of one tile:
Each tile is a square with side length 1 foot. Therefore, the area of one tile is:
$\text{Area of one tile} = 1 \times 1 = 1 \text{ square foot}$

2. Calculate the area of the white quarter circles on one tile:
Each tile has four quarter circles, each with a radius of $\frac{1}{2}$ foot. The area of one quarter circle is:
$\text{Area of one quarter circle} = \frac{1}{4} \pi \left( \frac{1}{2} \right)^2 = \frac{1}{4} \pi \left( \frac{1}{4} \right) = \frac{\pi}{16} \text{ square feet}$
Since there are four quarter circles on each tile, the total area of the white regions on one tile is:
$\text{Total area of white regions on one tile} = 4 \times \frac{\pi}{16} = \frac{4\pi}{16} = \frac{\pi}{4} \text{ square feet}$

3. Calculate the shaded area on one tile:
The shaded area on one tile is the total area of the tile minus the area of the white regions:
$\text{Shaded area on one tile} = 1 - \frac{\pi}{4} \text{ square feet}$

4. Calculate the total number of tiles:
The floor is 8 feet by 10 feet, so the total number of 1-foot by 1-foot tiles is:
$\text{Total number of tiles} = 8 \times 10 = 80$

5. Calculate the total shaded area on the floor:
The total shaded area on the floor is the shaded area on one tile multiplied by the total number of tiles:
$\text{Total shaded area} = 80 \times \left( 1 - \frac{\pi}{4} \right) = 80 - 80 \times \frac{\pi}{4} = 80 - 20\pi \text{ square feet}$

The final answer is $\boxed{80 - 20\pi}$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.