Final answer:
To calculate the position of the object in front of the concave mirror, we can use the mirror equation. By setting the image distance to be three times the object distance, we can solve for the object distance algebraically. The object should be positioned at -5 cm in front of the concave mirror.
Step-by-step explanation:
To find the position of the object in front of the concave mirror, we can use the mirror equation:
1/f = 1/v - 1/u
Where f is the focal length, v is the image distance, and u is the object distance.
Given that the focal length of the mirror is 30 cm and we want the image to be three times the size of the object, we can set v/u = 3.
Substituting the values into the mirror equation:
1/30 = 1/v - 1/(3u)
Simplifying the equation:
1/v = 1/30 + 1/(3u)
To solve for u, we can assume an arbitrary value for v and find the corresponding value of u.
Let's assume v = 60 cm:
1/60 = 1/30 + 1/(3u)
Simplifying the equation:
1/60 - 1/30 = 1/(3u)
Combining the terms:
1/60 = 2/60 + 1/(3u)
Combining the fractions on the right side:
1/60 = 3/60 + 1/(3u)
Combining the terms:
1/60 = 4/60 + 1/(3u)
Combining the fractions on the right side:
1/60 = 5/60 + 1/(3u)
Simplifying the equation:
1/60 - 5/60 = 1/(3u)
Combining the terms:
-4/60 = 1/(3u)
Dividing both sides by -4/180:
-1/15 = 1/(3u)
Multiplying both sides by 3u:
-3u/15 = 1
Dividing both sides by -3/15:
u = -5
The object should be positioned at -5 cm in front of the concave mirror.