You have a painting that is 30 in. wide and 22.5 in. tall. You would like to reproduce it on a sheet of paper that measures 8 1/2 in. by 11 in., leaving at least a 1in. margin on all four sides. a) What scale should you use if you keep the sheet of paper in the normal upright orientation? Assume that the reproduction will be as large as possible. b) What scale should you use if you turn the paper on its side?
A 1-inch margin on all four sides of an 8 1/2 in. by 11 in. piece of paper leaves a 6 1/2 in. by 9 in. piece of paper to work with.
a) The painting is in landscape orientation, 30 in. wide by 22.5 in. tall.
The usable paper is in portrait orientation, 6 1/2 in. wide by 9 in. tall.
First, let's assume that the 22.5 in. measurement will be shrunken to exactly 9 in. and, keeping the aspect ratio constant, the 30 in. measurement must then be shrunken to some unknown length, x. We then have:
x/30 = 9/22.5
x = 12
Hmm...12 is too large, because we have only 6.5 in. to work with in that direction.
So, assume instead that the 30 in. measurement will be shrunken to exactly 6.5 in. and, keeping the aspect ratio constant, the 22.5 in. measurement must then be shrunken to some unknown length, y. We then have:
y/22.5 = 6.5/30
y = 39/8 or 4 7/8
This will fit on the paper, because we have 9 in. to work with in that direction.
So, a reduction scale of 6.5/30 works best, properly written as 13/60.
b) The painting is in landscape orientation, 30 in. wide by 22.5 in. tall.
The usable paper is also in landscape orientation, 9 in. wide by 6.5 in. tall.
Use a similar strategy to work out the best scale to use for this situation.
Thank you, Candlelight.
a) To determine the scale for reproducing the painting on a sheet of paper in the normal upright orientation, we need to calculate the maximum dimensions of the painting that can fit within the given paper size while leaving a 1-inch margin on all four sides.
First, we subtract the margins from the paper size:
Length of usable space = 8 1/2 in - (2 * 1 in) = 8 1/2 in - 2 in = 6 1/2 in
Width of usable space = 11 in - (2 * 1 in) = 11 in - 2 in = 9 in
Now we find the scale for both the height and width dimensions by dividing the usable space by the corresponding dimensions of the painting:
Scale for height = (usable height) / (painting height) = 6 1/2 in / 22.5 in
Scale for width = (usable width) / (painting width) = 9 in / 30 in
The smaller of the two scales will determine the scale to use to fit the painting on the paper, while leaving at least a 1-inch margin on all four sides. Let's calculate the scales:
Scale for height = (6 1/2 in) / (22.5 in) = 0.289
Scale for width = (9 in) / (30 in) = 0.3
Therefore, the scale to reproduce the painting on a sheet of paper in the normal upright orientation is 0.289.
b) To determine the scale for reproducing the painting on a sheet of paper when it is turned on its side, we exchange the values for the length and width of the usable space:
Length of usable space = 11 in - (2 * 1 in) = 11 in - 2 in = 9 in
Width of usable space = 8 1/2 in - (2 * 1 in) = 8 1/2 in - 2 in = 6 1/2 in
Now we calculate the scales for height and width using the new dimensions:
Scale for height = (usable height) / (painting height) = 9 in / 22.5 in
Scale for width = (usable width) / (painting width) = 6 1/2 in / 30 in
Again, we choose the smaller scale:
Scale for height = (9 in) / (22.5 in) = 0.4
Scale for width = (6 1/2 in) / (30 in) = 0.2167
Therefore, the scale to reproduce the painting on a sheet of paper when it is turned on its side is 0.2167.
a) To determine the scale for reproducing the painting on the sheet of paper in the normal upright orientation, we need to consider the maximum size of the reproduction while leaving a 1-inch margin on all sides.
Step 1: Calculate the available space on the sheet of paper after accounting for the margins.
The width of the paper after leaving a 1-inch margin on both sides is:
8.5 inches - 2 inches (1 inch margin on each side) = 6.5 inches.
The height of the paper after leaving a 1-inch margin at the top and bottom is:
11 inches - 2 inches (1 inch margin on each side) = 9 inches.
Step 2: Calculate the aspect ratio of the painting and the available space on the paper.
The aspect ratio of the painting is:
Width / Height = 30 inches / 22.5 inches = 4:3.
The aspect ratio of the available space on the paper is:
Width / Height = 6.5 inches / 9 inches = 13:18.
Step 3: Determine the scale factor by comparing the aspect ratios.
To keep the proportions of the painting intact, the aspect ratios of the painting and the available space on the paper should be the same.
Divide the aspect ratio of the available space on the paper by the aspect ratio of the painting:
13:18 ÷ 4:3 = (13/4) : (18/3) = 13/4 : 6 = 2.625 : 1.
Therefore, the scale factor for reproducing the painting on the sheet of paper in the normal upright orientation is 2.625 : 1.
b) To determine the scale for reproducing the painting on the sheet of paper when turned on its side, we need to repeat the steps with the dimensions of the paper swapped.
Step 1: Calculate the available space on the sheet of paper after accounting for the margins.
The width of the paper after leaving a 1-inch margin at the top and bottom is:
11 inches - 2 inches (1 inch margin on each side) = 9 inches.
The height of the paper after leaving a 1-inch margin on both sides is:
8.5 inches - 2 inches (1 inch margin on each side) = 6.5 inches.
Step 2: Calculate the aspect ratio of the painting and the available space on the paper.
The aspect ratio of the painting is:
Width / Height = 30 inches / 22.5 inches = 4:3.
The aspect ratio of the available space on the paper is:
Width / Height = 9 inches / 6.5 inches = 18:13.
Step 3: Determine the scale factor by comparing the aspect ratios.
Divide the aspect ratio of the available space on the paper by the aspect ratio of the painting:
18:13 ÷ 4:3 = (18/4) : (13/3) = 9/2 : 39/13 = 4.5/1 : 3.
Therefore, the scale factor for reproducing the painting on the sheet of paper when turned on its side is 4.5 : 3.