The circle in the figure below has a radius of r and center at C . The distance from A to B is x , the distance from A to D is y , and the length of arc BD is s . For the following problem, redraw the figure, label it with the values given, and then solve. If and , find x . Apply the rules regarding the use of significant digits after all computations are complete.

Question

The circle in the figure below has a radius of r and center at C . The distance from A to B is x , the distance from A to D is y , and the length of arc BD is s . For the following problem, redraw the figure, label it with the values given, and then solve. ​If   and  , find x . Apply the rules regarding the use of significant digits after all computations are complete.  ​

Quizplus · Accepted Answer

Draw the figure as instructed with values given:
[asy]
draw(circle((0,0),5));
pair C=(0,0);
dot(C);
pair A=(-3,0);
dot(A);
pair B=(3,0);
dot(B);
pair D=(-4,0);
dot(D);
draw(A--B);
draw(C--A);
draw(C--B);
draw(C--D);
label("r", (0,0), N);
label("x", (0,0)--(B), N);
label("$\frac{s}{2r}r$",(0,0),SSE);
label("y", (0,0)--(D), N);
[/asy]
By definition of the circle, arc BD has length $\frac{s}{2r}$ times the circumference of the circle, which is $2\pi r$. So we get equations:
\begin{align*}
\frac{s}{2r}(2\pi r)&=s \
\frac{s}{2r}(2\pi r)&=\frac{s}{2} .
\end{align*}
Since AD=r-y and AB=r+x, we obtain another equation:
\begin{align*}
r-y+x&=\frac{s}{2\pi}r
\end{align*}
Substituting, we have
\begin{align*}
\frac{s^2}{2r}&=\left(\frac{s}{2\pi}ight)^2 r^2-x^2+2xr-2ry+y^2 \
2s^2 &= 2\left(\frac{s}{\pi}ight)^2r^3 - 4r^2y + 2ry^2 + 2r^2x^2 - 4rx^2 \
2s^2 &= 2\left(\frac{s}{\pi}ight)^2r^3 - 4rx^2 - 4r^2y + 2ry^2 + 2r^2x^2 \
\end{align*}
Since the options vary significantly, we can make reasonable approximations of the given values to solve for x. 
$r \approx 5.4$, $y \approx 4.4$, $s = 3.3$,  $\frac{s}{\pi}\approx 1.05$
\begin{align*}
2(3.3^2) &= 2(1.05^2)(5.4^3) - 4(5.4)(x^2) - 4(5.4^2)(4.4) + 2(4.4)(y^2) + 2(5.4^2)(x^2) \
21.78 &= 324.69 - 23.2656x^2 - 1026.24 + 38.61 + 58.32x^2 \
21.78 &= - 967.63 - 23.2656x^2 + 58.32x^2 \
989.41&= 81.0556(x^2) \
12.21 &\approx (x^2) \
x &\approx 3.5
\end{align*}
Therefore, the best choice with the correct approximation is $\boxed{	extbf{(B)}}$.

The Circle in the Figure Below Has a Radius of R