Find the area of a square with side a when a=2 in, a=12 in, and a=0.8 in. Are the area of a square and the length of its side directly proportional quantities? Why or why not?
surely you know that the area of a square of side a is just A=a^2
To be directly proportional, you must have area A
A = ka, for some constant k
but this is not true.
To find the area of a square, you need to multiply the length of its side by itself (squared). Let's calculate the area for each value of side length (a) given:
For a = 2 in:
Area = a^2 = 2^2 = 4 square inches
For a = 12 in:
Area = a^2 = 12^2 = 144 square inches
For a = 0.8 in:
Area = a^2 = 0.8^2 = 0.64 square inches
Now let's analyze whether the area of a square and the length of its side are directly proportional or not.
Two quantities are directly proportional when a change in one quantity results in a corresponding proportional change in the other quantity. In the case of a square, since the area is calculated by squaring the side length, we can say that the area and the length of the side are directly proportional.
When you compare the calculations we just did, you can observe that as the side length increases, the area of the square also increases, and vice versa. For example, doubling the side length (from 2 in to 4 in) quadruples the area (from 4 square inches to 16 square inches). This consistent relationship indicates that the area and the length of the side are directly proportional quantities in a square.