An object has an angular size of 0.0140 rad when placed at the near point (20.0 cm) of an eye. When the eye views this object using a magnifying glass, the largest possible angular size of the image is 0.0490 rad. What is the focal length of the magnifying glass?

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To find the focal length of the magnifying glass, we can use the formula for magnification:

magnification = angular size of image / angular size of object

Given that the largest possible angular size of the image is 0.0490 rad and the angular size of the object is 0.0140 rad, we can write the equation:

magnification = 0.0490 rad / 0.0140 rad

Simplifying the equation gives us:

magnification = 3.5

We know that the magnification for a magnifying glass is given by the formula:

magnification = 1 + (1 / lens power)

To find the lens power, we rearrange the formula:

1 / lens power = magnification - 1

Simplifying further:

lens power = 1 / (magnification - 1)

Plugging in the value for magnification, we have:

lens power = 1 / (3.5 - 1)

Calculating the lens power gives us:

lens power = 1 / 2.5

lens power = 0.4 diopters

The focal length of a lens is given by the formula:

Focal length (in meters) = 1 / lens power (in diopters)

Converting the lens power to meters, we have:

Focal length (in meters) = 1 / 0.4 diopters

Calculating the focal length gives us:

Focal length (in meters) = 2.5 meters

Therefore, the focal length of the magnifying glass is 2.5 meters.