Solution
For a point on the sphere to be visible in the plane mirror below it, the light ray tangent to the sphere at that point must strike the mirror surface. If the mirror is smaller than the intersection point of this tangent, the ray will not be reflected.
1. Geometry Setup:
Let the sphere have radius $r$. The center is at $(0, r)$ and the mirror is on the x-axis ($y=0$).
The coordinates of the point $P$ at latitude $\theta$ are: $$ P(x, y) = (r \cos \theta, r + r \sin \theta) $$
Let the sphere have radius $r$. The center is at $(0, r)$ and the mirror is on the x-axis ($y=0$).
The coordinates of the point $P$ at latitude $\theta$ are: $$ P(x, y) = (r \cos \theta, r + r \sin \theta) $$
2. Equation of Tangent:
The slope of the radius vector to $P$ is $\tan \theta$ (with respect to horizontal at center). Actually, measuring latitude from the equator means the angle with the horizontal plane is $\theta$.
Slope of tangent $m = \tan(90^\circ + \theta) = -\cot \theta$.
Line Equation: $$ y – y_p = -\cot \theta (x – x_p) $$
The slope of the radius vector to $P$ is $\tan \theta$ (with respect to horizontal at center). Actually, measuring latitude from the equator means the angle with the horizontal plane is $\theta$.
Slope of tangent $m = \tan(90^\circ + \theta) = -\cot \theta$.
Line Equation: $$ y – y_p = -\cot \theta (x – x_p) $$
3. Finding the Mirror Radius (R):
We find the x-intercept (where $y=0$). Set $x = R$: $$ 0 – r(1+\sin\theta) = -\frac{1}{\tan\theta}(R – r\cos\theta) $$ $$ r(1+\sin\theta)\tan\theta = R – r\cos\theta $$ $$ R = r[\cos\theta + (1+\sin\theta)\tan\theta] $$
We find the x-intercept (where $y=0$). Set $x = R$: $$ 0 – r(1+\sin\theta) = -\frac{1}{\tan\theta}(R – r\cos\theta) $$ $$ r(1+\sin\theta)\tan\theta = R – r\cos\theta $$ $$ R = r[\cos\theta + (1+\sin\theta)\tan\theta] $$
4. Substitution:
Given $r=10$ cm, $\theta=37^\circ$. ($\sin 37^\circ=0.6, \cos 37^\circ=0.8, \tan 37^\circ=0.75$) $$ R = 10 [0.8 + (1.6)(0.75)] $$ $$ R = 10 [0.8 + 1.2] = 20 \text{ cm} $$
Given $r=10$ cm, $\theta=37^\circ$. ($\sin 37^\circ=0.6, \cos 37^\circ=0.8, \tan 37^\circ=0.75$) $$ R = 10 [0.8 + (1.6)(0.75)] $$ $$ R = 10 [0.8 + 1.2] = 20 \text{ cm} $$
Answer: 20 cm