1. Interpret the geometry

A football (sphere) of radius $R$ rests over a circular hole of radius $r$.

Let:

• $O$ = center of the sphere,
• $P$ = point on the rim of the hole where the sphere touches,
• the vertical axis pass through the center of the hole.

Since the sphere touches the rim, the distance from the center of the sphere to the rim point is the radius of the sphere:

$OP = R$

The horizontal distance of the rim point from the vertical axis is the radius of the hole:

$ext{horizontal distance} = r$

2. Form the right triangle

Consider the right triangle formed by:

• hypotenuse $OP = R$,
• opposite side to angle $\theta$ equal to $r$.

If $\theta$ is the angle made by $OP$ with the vertical, then

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{r}{R}$

3. Match with the options

Thus,

$\sin\theta = \frac{r}{R}$

So the correct option is:

A. $\sin\theta = \dfrac{r}{R}$

4. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They match.