There is a concave mirror of radius of curvature 30 cm. A U-shaped wire of shown dimensions is placed in front of concave mirror as shown in given figure. Find the total length of the image formed.

Explanation

Focal length of the concave mirror is half of its radius of curvature (R = 30 cm ) . And as per sign convention focal length of mirror can be written as follows :
f = -15 cm
Let us first solve for part AB using mirror formula
`(1)/(f) = (1)/(v) + (1)/(u)`
`implies " " (1)/((-15)) = (1)/(v) + (1)/((-45))`
`implies " " (1)/(v) = (1)/(45) - (1)/(15)`
`implies " " (1)/(v) = (1- 3)/(45)`
`implies " " v = -22.5` cm
Now , we can use formula of magnification.
`(h_(2))/(h_(1)) = - (v)/(u)`
`implies " " (h_(2))/(15) = ((-22.5))/((-45))`
`implies h_(2) = -7.5` cm
Negative means image is inverted and its height is 7.5 cm .
A.B. of height 7.5 cm shown in diagram is image of AB .
Let us now solve for part CD using mirror formula .
`(1)/(f) = (1)/(v) + (1)/(u)`
`implies " " (1)/((-15)) = (1)/(v) + (1)/((-60))`
`implies " " (1)/(v) = (1)/(60) - (1)/(15)`
`implies " " (1)/(v) = (1-4)/(60)`
`implies v = -20` cm
Now we can use formula of magnification .
`(h_(2))/(h_(1)) = - (v)/(u)`
`implies " " (h_(2))/(15) = - ((-20))/((-60))`
`implies " " h_(2) = - 5` cm
Negative means image is inverted and its height is 5 cm . CD. of height 5 cm shown in diagram is image of CD.

From the above figure we can calculate the length of image as follows :
Length of image = `A B. + B.C . + C.D.`
`implies` Length of image = 7.5 cm + 2.5 cm + 5 cm
`implies` Length of image = 15 cm .