Difference between revisions of "AoPS Wiki:Sandbox"

Revision as of 20:41, 2 April 2014

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Solution

We want to find the area of this figure:

$[asy] //import graph; draw(circle((0,0), 1)); draw(circle((1,0), 1)); [/asy]$

We want to find the area of this figure:

$[asy] path rt,tt, tri; real x, y; y = 1+sqrt(2); x = y+(6/1.7); tt=(0,0)..(y,1)--(y,-1)..cycle; rt=(y,-1)--(x,-1)--(x,1)--(y,1); tri=(y,-1)--(y-1,0)--(y,1); draw(rt); draw(tt); draw(tri); label("1.7", (x, 0), E); label("3", (y+(3/1.7), -1), S); label("C", (y-1, -0.1), S); [/asy]$

We label the circle as circle C. We can break the figure into three parts, shown as the 3/4 circle, the triangle, and the rectangle.

Lets first take a look at the rectangle.

$[asy] path rt; real x, y; y = 1+sqrt(2); x = y+(6/1.7); rt=(y,-1)--(x,-1)--(x,1)--(y,1)--cycle; draw(rt); label("1.7", (x, 0), E); label("3", (y+(3/1.7), -1), S); [/asy]$

It has an area of $3 * 1.7 = 5.1$ .

Lets now take a look at the triangle, after drawing the height.

$[asy] unitsize(0.8inch); path tri, lin; real x, y; y = 1+sqrt(2); x = y+(6/1.7); tri=(y,-1)--(y-1,0)--(y,1)--cycle; lin=(y-1, 0)--(y,0); draw(tri); draw(lin); label("1.7", (y, 0), E); label("C", (y-1, -0.1), S); [/asy]$

We see that both the radii are the two shorter sides of the triangle, making this a isosceles 45-45-90 triangle.

We also see that the height that we drew is half the hypotenuse(note the two smaller 45-45-90 isosceles triangles).

Hence, the area of the triangle is $\frac{1.7 * \frac{1.7}{2}}{2} = 0.7225$ .

Now, let's take a look at the 3/4 circle. We know it is 3/4 because there is a 90 degree triangle cut out of it.

$[asy] unitsize(0.6inch); path tt, tri; real x, y; y = 1+sqrt(2); x = y+(6/1.7); tt=(0,0)..(y,1)--(y-1, 0)--(y,-1)..cycle; tri=(y, 1)--(y,-1); draw(tt); fill(tt, gray(0.4)); label("1.7", (y, 0), E); label("C", (y-1, -0.1), S); [/asy]$

We find the radius using the 45-45-90 triangle. Since the ratios of the sides are 1:1: $\sqrt{2}$ , we can find the radius to be $\frac{1.7}{\sqrt{2}} = \frac{1.7 \sqrt{2}}{2}$ .

Hence, the area of the whole circle is $\pi r^2$ , and the area of the 3/4 circle is $\frac{3}{4} * \pi * (\frac{1.7 \sqrt{2}}{2})^2$ .

Adding it all up, we find the answer to be $\frac{3}{4} \pi (\frac{1.7 \sqrt{2}}{2})^2 + 0.7225 + 5.1$ .

Just plug it into your calculator. And please understand where all the numbers came from before writing the answer down.

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Difference between revisions of "AoPS Wiki:Sandbox"

Revision as of 20:41, 2 April 2014

Solution

@@ Line 2: / Line 2: @@
 ==Solution==
+We want to find the area of this figure:
+<asy>
+//import graph;
+draw(circle((0,0), 1));
+draw(circle((1,0), 1));
+</asy>
+-------------------------
 We want to find the area of this figure: