A concave spherical mirror has a radius of curvature 15 cm. A 1 cm tall bulb is placed in front of the mirror such that its image is formed 10 cm in front of the mirror. If you measure distance from the optic center of the mirror,

(A) What is the focal length of the mirror?
(B) What is the position of the object?
(C) What is the magnification of the image?
(D) What is the height of the image?

a) 7.5 cm

b) -30.0cm
c) -0.33 cm
d) -0.33 cm
the negative sign indicates that the image is inverted

convex and divergent

To find the answers to these questions, we can use the mirror formula and the magnification formula. The mirror formula is given by:

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.

The magnification formula is given by:

magnification = height of image / height of object.

Let's solve each part step-by-step.

(A) To find the focal length of the mirror, we need to substitute the given values into the mirror formula. From the mirror formula, we have:

1/f = 1/v - 1/u.

Given that the radius of curvature is 15 cm, which means the focal length is half the radius of curvature, f = 15/2 = 7.5 cm. Also, the image distance is v = -10 cm (negative sign indicates a virtual image), and the object distance is u = -15 cm (negative sign indicates a real object).

Substituting these values, we get:

1/7.5 = 1/-10 - 1/-15.

Simplifying, we get:

1/7.5 = (3 - 2)/(-30).

1/7.5 = 1/(-30).

Therefore, the focal length of the mirror is f = 7.5 cm.

(B) To find the position of the object, we already have the object distance, which is given as u = -15 cm. The negative sign indicates that the object is placed on the same side as the light source. Thus, the position of the object is 15 cm in front of the mirror.

(C) To find the magnification of the image, we need to use the magnification formula:

magnification = height of image / height of object.

Given that the height of the object is 1 cm, and the height of the image is h, we can find the magnification by substituting these values:

magnification = h / 1.

We need to find the value of h.

(D) To find the height of the image, we can use the mirror formula:

1/f = 1/v - 1/u.

Substituting the given values, we get:

1/7.5 = 1/-10 - 1/-15.

Simplifying, we have:

1/7.5 = (3 - 2)/(-30).

1/7.5 = 1/(-30).

From this equation, we can find the value of v, which is the image distance.

v = -30 cm.

Now, we can use the magnification formula:

magnification = height of image / height of object.

Substituting the values of h = height of image and height of object = 1 cm, we get:

magnification = h / 1.

Therefore, the magnification of the image is equal to the height of the image, h.

To find the height of the image, we need to determine the value of h. We can use the magnification formula:

magnification = height of image / height of object.

Given that the magnification is equal to 1 (since we obtained the magnification as h by dividing both sides of the magnification formula by 1 cm), we have:

1 = h / 1.

Therefore, the height of the image is h = 1 cm.

In summary:
(A) The focal length of the mirror is 7.5 cm.
(B) The position of the object is 15 cm in front of the mirror.
(C) The magnification of the image is 1.
(D) The height of the image is 1 cm.

To find the answers to these questions, we can use the mirror formula and magnification formula.

First, let's define the given variables:
- Radius of curvature (R) = 15 cm
- Height of the object (h) = 1 cm
- Object distance (u) = ?
- Image distance (v) = -10 cm (since the image is formed in front of the mirror)
- Focal length (f) = ?
- Height of the image (h') = ?
- Magnification (m) = ?

(A) To find the focal length of the mirror (f), we can use the mirror formula:
1/f = 1/v - 1/u.

However, we only have the image distance (v) and need to find the object distance (u). We know that the image formed is 10 cm in front of the mirror, so the object distance can be determined as follows:
u = v + d
where d is the distance of the object from the optic center of the mirror.

Now, the optic center of the mirror is the midpoint of the radius of curvature. Thus, the distance from the optic center to the object is:
d = R + u.

Plugging in the given values:
d = 15 cm + u.

Since we want to measure distances from the optic center of the mirror, the object distance (u) is the sum of the object distance (u') from the mirror and the previously calculated distance (d):
u = u' + d.

With u = -10 cm (for the image distance) and d = 15 cm + u, we can solve for u by substituting the values:
u = -10 cm + (15 cm - 10 cm) = -5 cm.

Now, we can use the mirror formula:
1/f = 1/v - 1/u.

Plugging in the values:
1/f = 1/-10 cm - 1/-5 cm.

Simplifying the equation:
1/f = -1/10 cm + 1/5 cm.

Combining the fractions:
1/f = -1/10 cm + 2/10 cm.

1/f = 1/10 cm.

Thus, the focal length (f) of the mirror is 10 cm.

(B) Now, to find the position of the object, we can use the lens formula:
1/f = 1/v - 1/u.

Rearranging the formula, we can solve for the object distance (u):
1/u = 1/f - 1/v.

Plugging in the known values:
1/u = 1/10 cm - 1/-10 cm.

Simplifying the equation:
1/u = 1/10 cm + 1/10 cm.

1/u = 2/10 cm.

Therefore, u = 10/2 = 5 cm.

So, the position of the object is 5 cm in front of the optic center of the mirror.

(C) To find the magnification of the image (m), we can use the magnification formula:
m = -v/u.

Plugging in the values obtained:
m = -(-10 cm)/5 cm.

Simplifying the equation:
m = 10/5.

Therefore, the magnification of the image is 2.

(D) Finally, to find the height of the image (h'), we can use the magnification formula:
m = h'/h.

Plugging in the known values:
2 = h'/1 cm.

Therefore, the height of the image is 2 cm.