Question

Let C be a circle with radius 2–√. Which of the following statements is true? The area of C is a rational number because it is the product of 2, a rational number, and π, an irrational number. The area of C is an irrational number because it is the product of 2, a rational number, and π, an irrational number. The circumference of C is a rational number because it is the product of 22–√, an irrational number, and π, an irrational number. The circumference of C is an irrational number because it is the product of 22–√, a rational number, and π, an irrational number.

Answer 1

The circumference of C is a rational number because it is the product of 22–√, an irrational number, and π, an irrational number.

Explanation:

The product of the numbers is a rational number. Though pi is an irrational number, the presence of the rational number cancels the irrationality of the numbers. The circumference is given by the following formula:

• Circumference =
  `\pi D`

Hence, the circumference of the circle is a rational number.

Answer 2

We are given radius = sqrt(2).

And sqrt(2) is an irrational number because we can not write it as a simplest fraction p/q.

Also π is an irrational number because we cannot write down a simple fraction that equals π.

We know area of a circle = π r^2 and

Circumference of a circle = 2π r.

If we square square root 2, we would get 2.

That is rational number.

But if we multiply an irrational number by a rational number, it would give an irrational number only.

Therefore, true statements are :

The area of C is an irrational number because it is the product of 2, a rational number, and π, an irrational number.

The circumference of C is an irrational number because it is the product of 2*2–√, a rational number, and π, an irrational number.