A child looks at his reflection in a spherical tree ornament 12.0 cm in diameter, and sees an image of his face that us reduced by one-fourth. How far is his face from the ornament?
a child Look at the reflecting Christmas tree ball that has a diameter of 9.0 cm and seas and image of her face that is half the real size how far is the child face from the ball...
To find the distance between the child's face and the ornament, we need to determine the distance between the center of the ornament and the image of his face. Let's break down the steps:
Step 1: Calculate the radius of the ornament
The diameter of the ornament is given as 12.0 cm. Since the diameter is twice the radius, we divide it by 2 to find the radius.
Radius = Diameter / 2 = 12.0 cm / 2 = 6.0 cm
Step 2: Calculate the size of the child's face
The child sees an image of his face that is reduced by one-fourth. This means the image size is three-fourths of the actual face size.
Let the actual face size be x cm.
Reduced face size = 3/4 * x = (3/4)x
Step 3: Calculate the distance between the child's face and the ornament
In a spherical mirror or ornament, the distance between the object and the mirror surface is equal to the distance between the image and the mirror surface. Let's assume this distance as d cm.
The distance between the object (child's face) and the center of the ornament is given as (radius + d).
So, the reduced image distance would be (radius - d).
Given that the reduced face size is (3/4)x, we can use similar triangles to set up a proportion:
(reduced face distance) / (reduced face size) = (actual face distance) / (actual face size)
(radius - d) / ((3/4)x) = (radius + d) / x
Step 4: Solve the equation for d
Cross-multiply to get rid of the fractions:
x(radius - d) = ((3/4)x)(radius + d)
x(radius - d) = (3/4)(x(radius + d))
xr - xd = (3/4)(xr + xd)
4xr - 4xd = 3xr + 3xd
xr - 4xd = 3xr
2xr = 4xd
2x = 4d
x = 2d
Step 5: Substitute the values and solve for d
x = 2d
(3/4)x = (3/4)(2d) = (3/2)d
(radius - d) / ((3/4)x) = (radius + d) / x
(6.0 - d) / ((3/2)d) = (6.0 + d) / d
Cross-multiply:
d/(6.0 - d) = ((3/2)d)/ (6.0 + d)
Now, solve the equation:
2d^2 = 3(6.0 - d)(6.0 + d)
2d^2 = 3(36 - d^2)
2d^2 = 108 - 3d^2
5d^2 = 108
d^2 = 108/5
d = sqrt(108/5)
d ≈ 4.87 cm
Therefore, the child's face is approximately 4.87 cm away from the ornament.
To find the distance between the child's face and the ornament, we need to first find the distance between the child's face and the image of his face in the ornament.
Let's denote the distance between the child's face and the image as d1, and the distance between the image and the ornament as d2.
Since the reflection in the ornament reduces the image by one-fourth, the ratio between the size of the image and the child's face is 1/4. This means that the diameter of the image is 1/4 of the diameter of the child's face.
Given that the diameter of the child's face is 12.0 cm, the diameter of the image is 12.0 cm * (1/4) = 3.0 cm.
Since the image is formed on the inner surface of the ornament, which is spherical, the distance between the image and the ornament (d2) is equal to the radius of the ornament.
The radius of the ornament is half the diameter, so it is 12.0 cm / 2 = 6.0 cm.
Now, we can find the distance between the child's face and the image (d1) by subtracting the diameter of the image from the diameter of the child's face and dividing it by 2.
d1 = (12.0 cm - 3.0 cm) / 2 = 9.0 cm / 2 = 4.5 cm
Therefore, the distance between the child's face and the ornament is 4.5 cm.
The radius of curvature of the ornament is 6.0 cm and the focal length is half that, 3.0 cm. Because if is convex, the focal length is considered negative.
For the image position,
1/do + 1/di = 1/f = -1/3.0
Since there is a demagnification of 1/4, di = -do/4
I used a minus sign because the image will be virtual, behind the reflecting surface.
1/do + 1/di = 1/do -4/do = -1/3
-3/do = -1/3
do = 9 cm.