Final answer:
The area of a circle with a radius of 2 feet involves π, an irrational number, making the area also irrational. The formula A = πr² yields an irrational result. For practical purposes, in calculations with significant figures, this area could be approximated and then rounded according to the least precise measurement.
Step-by-step explanation:
When finding the area of a circle with a radius of 2 feet, the area in question will involve using the value of π (pi), which is an irrational number. As π cannot be expressed as a fraction of two integers, it has a non-repeating, non-terminating decimal expansion. Hence, since the area of a circle is given by the formula A = πr², where r stands for radius, when you plug in the radius of 2 feet into this formula, the calculation will yield π multiplied by 4 square feet, which is an irrational number because it involves π. Despite being able to approximate π to a certain number of decimal places or significant figures for practical applications, its true value remains irrational.
For example, if a calculator with an eight-digit precision is used, and only two significant figures for the radius are considered, the calculated area could look something like this: A = πr² = (3.1415927...) x (2 feet)² = 12.5663704 square feet. However, due to the rules of significant figures, the result should be rounded to maintain the same level of precision as the least precise measurement involved in the calculation, which, in this case, is the radius with two significant figures. So, one would report the area as A = 13 square feet (rounded to two significant figures).