The image behind a convex mirror (radius of curvature = 71 cm) is located 24 cm from the mirror.
(a) Where is the object located?
The focal length of the mirror is
f = -R/2 = -35.5 cm
The image location is
Di = -24 cm,
since it behind the mirror.
Use the basic equation:
1/Do + 1/Di = 1/f = -1/35.5
1/Do = 1/24 - 1/35.5
Solve for object distance Do
Well, if the image is behind a convex mirror, that means it's a virtual image. So, to find the location of the object, we can use the mirror equation:
1/f = 1/do + 1/di
Where f is the focal length of the mirror, do is the object distance, and di is the image distance.
Since the mirror is convex, the focal length (f) will be positive. And in this case, the focal length is half the radius of curvature (R), so f = R/2.
Given that the radius of curvature is 71 cm, the focal length will be 71/2 = 35.5 cm.
And we know that the image distance (di) is 24 cm (since the image is located 24 cm from the mirror).
So, let's solve the mirror equation for the object distance (do):
1/35.5 = 1/do + 1/24
To find the object distance, I'll use my trusty calculator... *beep boop beep boop* ...and I calculated that the object distance is approximately 45.72 cm.
So, the object is located approximately 45.72 cm away from the convex mirror.
To determine the object's location, we can use the mirror equation:
1/f = 1/di + 1/do
Where:
- f is the focal length of the convex mirror.
- di is the image distance (distance of the image from the mirror).
- do is the object distance (distance of the object from the mirror).
For a convex mirror, the focal length (f) is negative.
Given:
- Radius of curvature (R) of the mirror is 71 cm.
- Image distance (di) is 24 cm.
To find the object distance (do), we need to calculate the focal length (f) first using the formula:
R = 2f
Substituting the given value of R (71 cm):
71 cm = 2f
Dividing both sides by 2:
f = 71 cm / 2 = 35.5 cm
Now that we have the focal length (f), we can substitute the values into the mirror equation and solve for do:
1/f = 1/di + 1/do
Substituting the known values:
1/35.5 cm = 1/24 cm + 1/do
Now, let's solve for do. First, simplify the equation:
1/35.5 cm - 1/24 cm = 1/do
To combine the fractions, find a common denominator:
(24 - 35.5) / (24 * 35.5) = 1/do
-11.5 / 852 = 1/do
Invert both sides to solve for do:
do = 852 / -11.5
do ≈ -73.91 cm
Therefore, the object is located approximately 73.91 cm in front of the convex mirror.
To find the location of the object in a convex mirror, we need to use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance from the mirror (negative value)
u = object distance from the mirror (negative value)
In this case, we are given:
Radius of curvature (R) = 71 cm
Image distance (v) = -24 cm
The relation between the radius of curvature and focal length is given by:
R = 2f
Therefore,
f = R/2 = 71 cm/2 = 35.5 cm
Substituting the values into the mirror formula:
1/35.5 = 1/(-24) - 1/u
Simplifying the equation further:
1/35.5 = (-1/24) - 1/u
To solve for u, we need to rearrange the equation:
1/35.5 + 1/24 = -1/u
Now, we can calculate it:
1/u = 1/35.5 + 1/24
Taking the reciprocal of both sides:
u = 1 / (1/35.5 + 1/24)
Using a calculator to add the fractions and then taking the reciprocal:
u ≈ 15.51 cm
Therefore, the object is located approximately 15.51 cm from the convex mirror.