Questions: The number t is rational and positive. Which statement about π · t is true? π · t is rational. π · t is irrational. π · t can be rational or irrational, depending on the value of t.

The number t is rational and positive. Which statement about π · t is true?
π · t is rational.
π · t is irrational.
π · t can be rational or irrational, depending on the value of t.

Transcript text: The number $t$ is rational and positive. Which statement about $\pi \cdot t$ is true? $\pi \cdot t$ is rational. $\pi \cdot t$ is irrational. $\pi \cdot t$ can be rational or irrational, depending on the value of $t$.

Solution

Solution Steps

To determine the nature of $\pi \cdot t$, we need to consider the properties of rational and irrational numbers. Since $\pi$ is an irrational number, the product of an irrational number and a non-zero rational number is always irrational. Therefore, $\pi \cdot t$ is irrational.

Step 1: Understand the Properties of $\pi$ and Rational Numbers

The number $\pi$ is known to be an irrational number. A rational number is any number that can be expressed as the quotient of two integers, whereas an irrational number cannot be expressed in such a form.

Step 2: Analyze the Product $\pi \cdot t$

Given that $t$ is a positive rational number, we need to determine the nature of the product $\pi \cdot t$. The product of an irrational number and a non-zero rational number is always irrational. This is because multiplying an irrational number by a rational number does not result in a number that can be expressed as a simple fraction of two integers.

Step 3: Conclusion Based on Mathematical Properties

Since $\pi$ is irrational and $t$ is rational, the product $\pi \cdot t$ is irrational. Therefore, the statement that $\pi \cdot t$ is irrational is true.