To solve this problem, we need to understand the concept of proportional relationships in the context of image projection. When you move a projector further away from the screen, the size of the projected image increases proportionally. This means that the dimensions of the image will change based on the ratio of the distances from the projector to the screen.
Identifying the Initial Conditions
Initially, the image is projected onto a screen that is 10 feet away, with dimensions of 2 feet by 3 feet. We can denote the initial distance as D1 = 10 feet and the dimensions of the image as W1 = 2 feet (width) and H1 = 3 feet (height).
Setting Up the Proportional Relationship
Now, when the screen is moved to a distance of 15 feet, we denote this new distance as D2 = 15 feet. We want to find the new dimensions of the image, which we will call W2 and H2.
Using the Proportionality Formula
The relationship between the dimensions of the image and the distances can be expressed as:
- W1 / D1 = W2 / D2
- H1 / D1 = H2 / D2
Calculating the New Width
Let's first calculate the new width:
Using the width formula:
W2 = W1 * (D2 / D1)
Substituting in the values:
W2 = 2 feet * (15 feet / 10 feet)
W2 = 2 feet * 1.5 = 3 feet
Calculating the New Height
Next, we calculate the new height:
Using the height formula:
H2 = H1 * (D2 / D1)
Substituting in the values:
H2 = 3 feet * (15 feet / 10 feet)
H2 = 3 feet * 1.5 = 4.5 feet
Final Dimensions of the Projected Image
After performing the calculations, we find that the new dimensions of the projected image when the screen is 15 feet away are:
Width: 3 feet and Height: 4.5 feet.
Choosing the Correct Option
Looking at the options provided:
- (A) 3 feet by 4.5 feet
- (B) 4 feet by 6 feet
- (C) 4.5 feet by 6.75 feet
- (D) 7 feet by 8 feet
The correct answer is (A) 3 feet by 4.5 feet.
