A dime is 1.8 cm in diameter. Earth's moon is 3.844´1010 cm from Earth and has a diameter of 3.76´108 cm. At what distance from your eye would you have to hold a dime so that it has the same angular diameter as the full moon?

2 inches

To determine the distance from your eye at which a dime would have the same angular diameter as the full moon, you can use the formula for angular diameter:

Angular Diameter = 2 * arctan( Diameter / (2 * Distance) )

Where:
- Angular Diameter is the angle that the object subtends at the eye.
- Diameter is the actual diameter of the object.
- Distance is the distance from the eye to the object.

Let's calculate the angular diameter of the full moon using its given diameter and distance from Earth.

Angular Diameter of Moon = 2 * arctan( Diameter of Moon / (2 * Distance to Moon) )

= 2 * arctan( 3.76´108 cm / (2 * 3.844´1010 cm) )

= 2 * arctan( 3.76´108 cm / 7.688´1010 cm )

= 2 * arctan( 4.886´10^-3 )

Now, we need to find the distance at which a dime would have the same angular diameter as the full moon.

Angular Diameter of Dime = 2 * arctan( Diameter of Dime / (2 * Distance from Eye) )

We know the diameter of the dime is 1.8 cm. Let's substitute the angular diameter of the full moon and solve for Distance from Eye.

2 * arctan( 1.8 cm / (2 * Distance from Eye) ) = 2 * arctan( 4.886´10^-3 )

Now, we need to solve for Distance from Eye.

arctan( 1.8 cm / (2 * Distance from Eye) ) = arctan( 4.886´10^-3 )

To isolate Distance from Eye, we can take the tangent of both sides.

tan( arctan( 1.8 cm / (2 * Distance from Eye) ) ) = tan( arctan( 4.886´10^-3 ) )

This simplifies to:

1.8 cm / (2 * Distance from Eye) = 4.886´10^-3

Now, we can solve for Distance from Eye.

Distance from Eye = 1.8 cm / (2 * 4.886´10^-3)

= 1.8 cm / 9.772´10^-3

= 184.67 cm

Therefore, you would need to hold a dime approximately 184.67 cm (or 1.85 meters) from your eye to have the same angular diameter as the full moon.