David wants to reduce
a rectangular image on his computer so that
the new image has 25% of the area of the
original image. How should David modify
the image? Justify your answer.
area = constant * length ^2
1/4 area = constant * (length/x)^2
x^2 = 4
x = 2
divide all lengths by 2
area scales with square of length ratio
if photo is 10 by 20, area - 200
divide lengths by 2
5 by 10 , area = 50
which is 1/4 of original area
To reduce the area of a rectangular image, David will need to modify both the length and width of the image while maintaining the same aspect ratio.
Let's say the original image has a length of L and a width of W. The area of the original image is given by A = L * W.
David wants the new image to have 25% of the area of the original image. So, the area of the new image should be 0.25 * A.
To achieve this, David needs to find the new length (L') and new width (W') such that L' * W' = 0.25 * A.
Let's solve for L' and W' by taking the square root of both sides:
√(L' * W') = √(0.25 * A)
√(L' * W') = √(0.25 * L * W)
√(L' * W') = 0.5 * √(L * W)
Therefore, the new length (L') and new width (W') should each be 0.5 * the square root of the original length (L) multiplied by the original width (W).
By modifying the image's length and width to be 0.5 * √(L * W), David will achieve an image that has 25% of the area of the original image.