A girl sees her image in a shiny glass sphere tree ornament that has a diameter of 18 cm. The image is upright and is located 2.9 cm behind the surface of the ornament. How far from the ornament is the child located?
The child is located cm in front of the ornament.
To find the distance of the child from the ornament, we can use the concept of the mirror formula:
1/f = 1/u + 1/v
where:
f = focal length of the mirror (in this case, the glass sphere ornament)
u = object distance (distance of the child from the ornament)
v = image distance (distance of the image from the ornament)
In this scenario, the image is located 2.9 cm behind the surface of the ornament (v = -2.9 cm). We need to solve for u.
Since the ornament is spherical, we can use the mirror formula for a convex mirror, where the focal length is half the radius of curvature:
f = R/2
The diameter of the ornament is given as 18 cm, so the radius of curvature (R) is half of that:
R = 18 cm / 2 = 9 cm
Substituting the value of R into the equation, we get:
f = 9 cm / 2 = 4.5 cm
Now we can rearrange the mirror formula to solve for u:
1/u = 1/f - 1/v
Substituting the known values:
1/u = 1/4.5 cm - 1/-2.9 cm
Simplifying:
1/u = (2.9 - 4.5) / (4.5 * -2.9) cm
1/u = -1.6 / -13.05 cm
1/u = 0.1226
To isolate u, we take the reciprocal of both sides:
u = 1 / 0.1226 cm
u ≈ 8.17 cm
Therefore, the child is located approximately 8.17 cm in front of the ornament.