KNOWNS:
Diverging (convex) mirror
Radius of curvature = 60 cm
(f = 30 cm)
M = 1/5
Hi = 20 cm
UNKNOWNS:
Ho?
Di?
Do?
EQUATIONS:
1/Do + 1/Di = 1/f
M = Hi/Ho = -Di/Do
I got:
Ho=100 cm
Di=-24 cm
Do=120 cm
I rearranged -Di/Do=1/5 to get -5Di=Do.
Then I substituted -5Di in for Do in 1/Do + 1/Di = 1/30 to solve for Di. Is that right?
(I'm not sure about the negative sign)
I doubt if it is right because you should be using -30 for the focal length, since it is a convex mirror.
To find the unknowns Ho, Di, and Do in this problem, we can use the given equations and known values. Let's go step by step.
1. Start with the equation relating the object distance (Do), image distance (Di), and focal length (f) of a mirror:
1/Do + 1/Di = 1/f [Equation 1]
2. Given that the radius of curvature (R) is 60 cm and f = 30 cm for a diverging (convex) mirror, we can use the relationship f = R/2 to calculate R.
R = 2f = 60 cm
3. Now, substitute the known values into Equation 1 and solve for Do:
1/Do + 1/Di = 1/f
1/Do + 1/Di = 1/(30 cm)
Since we are looking for Ho, which corresponds to the object height, we assume that the height of the object, Hi, is positive.
Now, rearrange Equation 1 to solve for Di:
1/Di = 1/f - 1/Do
1/Di = 1/30 cm - 1/Do
4. Now, consider the magnification equation:
M = Hi/Ho = -Di/Do [Equation 2]
Given M = 1/5 and Hi = 20 cm, we can substitute these values into Equation 2 and solve for Ho:
1/5 = 20/Ho
Ho = 100 cm
5. Now, substitute Ho = 100 cm into Equation 1 to find Di:
1/Di = 1/30 cm - 1/Do
1/Di = 1/30 cm - 1/(100 cm)
1/Di = 1/30 cm - 1/100 cm
1/Di = (10 - 3)/300 cm
1/Di = 7/300 cm
Di = 300 cm/7
6. Finally, substitute Ho = 100 cm and Di = 300 cm/7 into Equation 2 to find Do:
-Di/Do = Hi/Ho
-(300 cm/7)/Do = 20/100
-(300 cm/7)/Do = 1/5
Cross-multiplying:
-(300 cm/7) = (1/5)Do
Do = -(7/300) * (5/1) cm
Do = -7/60 cm
Since object distance (Do) is negative, it means the object is located on the same side as the observer, indicating a virtual image.
To summarize the results:
Ho = 100 cm (object height)
Di = 300 cm/7 (image distance)
Do = -7/60 cm (object distance)