Difference between revisions of "2013 UNCO Math Contest II Problems/Problem 1"

(Solution)
(Solution)
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== Solution ==
 
== Solution ==
<math>48-9\pi</math>
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Notice that the shaded area is equivalent to the area of a rectangle subtracted by the area of 2 semicircles. Since we know that the radius of the circles are <math>3</math>, we can find the width of the rectangle to be <math>6</math>. Solving for the shaded area, we get
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<cmath>
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\begin{align*}
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(8)(6) - 2\left(\dfrac12 \cdot 3^2\pi\right) &= 48 - 2\left(\dfrac12 \cdot 9\pi\right) \
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&=\boxed{48-9\pi}.
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\end{align*}
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</cmath>
  
 
== See Also ==
 
== See Also ==

Revision as of 13:12, 24 February 2022

Problem

[asy] filldraw((-4,-3)--(-4,3)--(4,3)--(4,-3)--cycle,grey); filldraw(circle((-4,0),3),white); filldraw(circle((4,0),3),white); draw((-7,3)--(7,3),black); draw((-7,-3)--(7,-3),black); [/asy]

In the diagram, the two circles are tangent to the two parallel lines. The distance between the centers of the circles is $8$, and both circles have radius $3$. What is the area of the shaded region between the circles?


Solution

Notice that the shaded area is equivalent to the area of a rectangle subtracted by the area of 2 semicircles. Since we know that the radius of the circles are $3$, we can find the width of the rectangle to be $6$. Solving for the shaded area, we get \begin{align*} (8)(6) - 2\left(\dfrac12 \cdot 3^2\pi\right) &= 48 - 2\left(\dfrac12 \cdot 9\pi\right) \\ &=\boxed{48-9\pi}. \end{align*}

See Also

2013 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions