Questions   
Sort: 
 #2
avatar+118723 
0
May 19, 2015
 #3
 #2
 #1
 #140
avatar
+3
May 19, 2015
 #6
avatar+26400 
+15

From the figure shown below, DE is the diameter of circle A and BC is the radius of circle B. If DE = 60 cm and AC = 10 cm, find the area of the shaded region

$$h=\overline{AC}=10~\rm{cm}
\qquad r_{A} = \frac{ \overline{DE} }{2}= 30~\rm{cm}
\qquad r_{B} = \overline{CB} \\\\
r_A^2 = h\cdot(2\cdot r_B-h)\qquad \Rightarrow\qquad
r_B = \frac{h+\dfrac{r_A^2}{h} }{2}=\frac {10+90}{2}= 50 ~\rm{cm}\\\\
\small{\text{Angle DBE $=B
\qquad B\ensurement{^{\circ}} = 2\cdot\arcsin{ \left( \dfrac{r_A}{r_B} \right)} $}} \\\\
\small{ A_{Blue}= \pi\cdot r_B^2 - \pi \frac{\cdot r_A^2}{2}
-\left[
\pi \cdot r_B^2 \cdot \frac{B\ensurement{^{\circ}} } { 360\ensurement{^{\circ}} } - \frac{1}{2}\cdot (2\cdot r_A)\cdot(r_B - h)
\right]
$}} \\\\
\small{ A_{Blue}= \pi\cdot r_B^2 - \pi \frac{\cdot r_A^2}{2}
-\left[
\pi \cdot r_B^2 \cdot \frac{ 2\cdot\arcsin{ \left( \dfrac{r_A}{r_B} \right)} } { 360\ensurement{^{\circ}} } - \frac{1}{2}\cdot (2\cdot r_A)\cdot(r_B - h)
\right]
$}} \\\\
\small{ A_{Blue}= \pi\cdot r_B^2 - \pi \frac{\cdot r_A^2}{2}
-\left[
\pi \cdot r_B^2 \cdot \frac{ \arcsin{ \left( \dfrac{r_A}{r_B} \right)} } { 180\ensurement{^{\circ}} } - r_A\cdot(r_B - h)
\right]
$}}$$

$$\small{ A_{Blue}= \pi\cdot r_B^2
\left(
1- \frac{ \arcsin{ \left( \dfrac{r_A}{r_B} \right)} } { 180\ensurement{^{\circ}} }
\right)
- \pi \frac{\cdot r_A^2}{2}+ r_A\cdot(r_B - h)
\right]
$}} \\\\
\small{ A_{Blue}= \pi\cdot 50^2
\left(
1- \frac{ \arcsin{ \left( \dfrac{30}{50} \right)} } { 180\ensurement{^{\circ}} }
\right)
- \pi \frac{\cdot 30^2}{2}+ 30\cdot(50 - 10)
\right]
$}} \\\\
\small{ A_{Blue}= \pi\cdot 50^2\cdot
0.7951672353008667 - \pi \frac{\cdot 30^2}{2}+ 1200\right]
$}} \\\\
\small{ A_{Blue}= 6031.5121678758663628914 ~\rm{cm^2}
$}}\\\\
A_{Blue} \approx 6031.5122 ~\rm{cm^2}$$

.
May 19, 2015

1 Online Users