A concave mirror is constructed so that a man at a distance of 26 from the mirror sees his image magnified 1.5 times. What is the radius of curvature of the mirror?
3. A concave shaving mirror is construction so that a man at a distance of 20cm from the mirror sees his image magnified 1.5 times. What is the radius of curvature of the mirror?
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To find the radius of curvature of the concave mirror, you can use the mirror formula, which states:
1/f = 1/v - 1/u,
where f is the focal length of the mirror, v is the image distance, and u is the object distance.
In this case, the man sees his image magnified 1.5 times, which means the magnification (m) is 1.5. The magnification is given by:
m = -v/u,
where the negative sign indicates that the image is virtual and magnified. Rearranging the equation, we have:
v/u = -m.
Substituting the given values, we get:
v/u = -1.5.
Now, we know that the object distance (u) is the distance of the man from the mirror, which is given as 26 cm. Let's substitute this value into the equation:
v/26 = -1.5.
To solve for v, we can multiply both sides of the equation by 26:
v = -1.5 * 26.
Calculating, we find that v is equal to -39 cm.
Now that we have the values for v and u, we can substitute them into the mirror formula:
1/f = 1/v - 1/u,
1/f = 1/-39 - 1/26.
Simplifying,
1/f = (-26 - 39) / (-39 * 26).
1/f = -65 / (-1014).
Now, take the reciprocal of both sides of the equation to solve for f:
f = -1014 / -65.
Calculating, we find that f is equal to approximately 15.6 cm.
The radius of curvature (R) of a concave mirror is twice the focal length (R = 2f). Therefore, the radius of curvature of the mirror is approximately 31.2 cm.