A movie projector is 6 ft away from the screen showing a rectangular picture that is 3 ft wide and 2 ft high.
A. Suppose the projector is moved to 12 ft from the screen. What size will the picture be (width, height, and area)? I figured 6 wide, 4 high, 24 area. Is this correct?
B. Suppose the projector is moved to a point 9 ft from the screen. What size will the picture be (width, hieght, and area? I'm confused on this one.
Many thanks!
a:w=6 h=4 a=24
A. To find the size of the picture when the projector is moved to 12 ft from the screen, you can use the concept of similar triangles.
The original setup with the projector 6 ft away from the screen and the picture being 3 ft wide and 2 ft high forms a right triangle. The base of the triangle is the distance from the projector to the screen (6 ft), and the height is the height of the picture (2 ft).
When the projector is moved to 12 ft away, the distance from the projector to the screen doubles. Let's call the width and height of the new picture W and H, respectively.
Using the concept of similar triangles, we can set up the following ratio:
(Width of original picture) / (Distance of original projector) = (Width of new picture) / (Distance of new projector)
Plugging in the values we know:
3 ft / 6 ft = W ft / 12 ft
Simplifying the equation gives:
1/2 = W / 12
Multiply both sides of the equation by 12:
6 = W
So, the width of the new picture is 6 ft.
Similarly, we can set up a ratio for the height:
(Height of original picture) / (Distance of original projector) = (Height of new picture) / (Distance of new projector)
Using the values we know:
2 ft / 6 ft = H ft / 12 ft
Simplifying the equation gives:
1/3 = H / 12
Multiply both sides of the equation by 12:
4 = H
So, the height of the new picture is 4 ft.
Finally, we can calculate the area of the new picture:
Area = Width x Height = 6 ft x 4 ft = 24 sq ft
Therefore, your initial assumption is correct. The size of the picture when the projector is moved to 12 ft from the screen will be 6 ft wide, 4 ft high, and have an area of 24 sq ft.
B. To find the size of the picture when the projector is moved to a point 9 ft from the screen, we can use the same approach as in part A.
Setting up the ratios:
(Width of original picture) / (Distance of original projector) = (Width of new picture) / (Distance of new projector)
3 ft / 6 ft = W ft / 9 ft
Simplifying the equation gives:
1/2 = W / 9
Multiply both sides of the equation by 9:
4.5 = W
So, the width of the new picture is 4.5 ft.
Setting up the ratio for the height:
(Height of original picture) / (Distance of original projector) = (Height of new picture) / (Distance of new projector)
2 ft / 6 ft = H ft / 9 ft
Simplifying the equation gives:
1/3 = H / 9
Multiply both sides of the equation by 9:
3 = H
So, the height of the new picture is 3 ft.
Finally, we can calculate the area of the new picture:
Area = Width x Height = 4.5 ft x 3 ft = 13.5 sq ft
Therefore, when the projector is moved to a point 9 ft from the screen, the size of the picture will be 4.5 ft wide, 3 ft high, and have an area of 13.5 sq ft.
I hope this helps! Let me know if you have any further questions.
To solve these questions, we can use the concept of similar triangles. Similar triangles have the same shape but different sizes. In this case, we can consider the original triangle formed by the projector, screen, and the picture, and compare it to the new triangle formed when the projector is moved.
Let's solve each part of the problem step by step:
A. When the projector is moved to 12 ft from the screen:
To determine the new size of the picture, we can set up a proportion between the original and new distances of the projector to the screen.
Original width / Original distance = New width / New distance
Plugging in the given values:
3 ft / 6 ft = New width / 12 ft
Simplifying the proportion:
(3/6) = (New width/12)
1/2 = New width/12
Cross-multiplying:
New width = (1/2) * 12
New width = 6 ft
Similarly, we can calculate the new height:
2 ft / 6 ft = New height / 12 ft
(2/6) = (New height/12)
1/3 = New height / 12
New height = (1/3) * 12
New height = 4 ft
Finally, to calculate the area:
Area = New width * New height
Area = 6 ft * 4 ft
Area = 24 square ft
So, your calculations are indeed correct for part A. The new size of the picture will be 6 ft wide, 4 ft high, with an area of 24 square ft.
Now, let's move on to part B:
B. When the projector is moved to a point 9 ft from the screen:
Similarly, we can set up the proportion:
3 ft / 6 ft = New width / 9 ft
Simplifying the proportion:
(3/6) = (New width/9)
1/2 = New width/9
Cross-multiplying:
New width = (1/2) * 9
New width = 4.5 ft
To calculate the new height, we use a similar proportion:
2 ft / 6 ft = New height / 9 ft
(2/6) = (New height/9)
1/3 = New height / 9
New height = (1/3) * 9
New height = 3 ft
For the area:
Area = New width * New height
Area = 4.5 ft * 3 ft
Area = 13.5 square ft
Therefore, the calculations for part B are as follows: The new size of the picture will be 4.5 ft wide, 3 ft high, with an area of 13.5 square ft.
I hope this explanation helps! Let me know if you have any further questions.