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# Geometry of circles and rectangles AMC 8 2014 problem 20

Try this beautiful problem from AMC 8. It involves geometry of circles and rectangles. We provide sequential hints so that you can try the problem.

# What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Geometry of circles and rectangles This problem from American Mathematics contest (AMC 8, 2014) will help us to learn more about geometry of circles and rectangles.

# Next understand the problem

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Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?$[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy]$[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.0.9″ open=”on”]American Mathematical Contest 2014, AMC 8 Problem 20

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Geometry of circles and rectangles [/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]6/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

[/et_pb_text][et_pb_tabs _builder_version=”4.1″][et_pb_tab title=”Hint 0″ _builder_version=”4.1″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”Hint 1″ _builder_version=”4.1″]The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle.[/et_pb_tab][et_pb_tab title=”Hint 2″ _builder_version=”4.1″]Here the area of the rectangle is 3.5=15. Area of quater circles is (Area of the circle )/4 = $\frac{\pi . r^2}{4}$ , where r= radius of the circle . so, The area of all 3 quarter circles is $\frac{\pi}{4}+\frac{\pi(2)^2}{4}+\frac{\pi(3)^2}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}$, where area of the quater for circle A is $\frac{\pi}{4}$ ,for circle B is $\frac {\pi .2^2}{4}$ , for circle C is $\frac{\pi.3^2}{4}$.Therefore the area in the rectangle but outside the circles is $15-\frac{7\pi}{2}$.[/et_pb_tab][et_pb_tab title=”Hint 3″ _builder_version=”4.1″]Now what can we do with  $15-\frac{7\pi}{2}$ to get an approximate value ?[/et_pb_tab][et_pb_tab title=”Hint 4 ” _builder_version=”4.1″]As we know that we can approximate $\pi$ by $\frac{22}{7}$ .  and substituting that in will give 15-11=4.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]