A child looks at his reflection in a spherical tree ornament 12 cm in diameter, and sees an image of his face that is reduced by one-fourth. How far is his face from the ornament?

I don't know what I would do with the given information. I know that M=-q/p and M=1/4. Any help is appreciated!

You know the radius of curvature for the sphere, so f = 1/2 radius curvature. Use the magnification relationship M= 1/4, then find di/do = 1/4, and finally, if di=.25do, put that in...

1/di + 1/do= 1/f and solve for do.

Can you check? Thanks!

M=-q/p

q=R/2=6cm/2=3cm

1/4=-3cm/p

p=-12cm

Yes.

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To solve this problem, we can use the concept of similar triangles.

We know that the child sees a reduced image of his face, which means that the ratio of the height of the image to the actual height of his face is 1/4.

Let's assume that the distance between the child's face and the ornament is "x." We can create a right triangle with the child's face as one side, the distance between his face and the ornament as the hypotenuse, and the image of his face as the other side.

Now, let's examine the similar triangles formed by this right triangle and the one formed by the child's face and its image:

1. The original triangle: The height of the child's face is the diameter of the ornament, which is 12 cm, and the hypotenuse is the distance between the child's face and the ornament, which is "x."

2. The image triangle: The height of the image is one-fourth of the child's face's height, which is 1/4 * 12 cm = 3 cm. The hypotenuse is the same as in the original triangle, which is "x."

Now, we can set up a proportion using the corresponding sides of the similar triangles:

(child's face height) / (image height) = (distance from face to ornament) / (distance from face to image)

12 cm / 3 cm = x / x

Simplifying the equation, we get:

4 = x / x

Since the x's cancel each other out, we are left with the equation:

4 = 1

This equation is not true, which means there is an error in our assumption or calculation.

Therefore, it is not possible to determine the distance between the child's face and the ornament based on the given information alone. There might be missing information or a mistake in the problem statement.