Question

5 multiple choice 10 points find the area of the rectangle shown. 6√10 3√2 9√12 18√20 12√10 + 6√2 36√5 6 multiple choice 10 points which number is irrational? √2 3/4 3.125 7.9 7 multiple choice 10 points if a is a rational number and b is an irrational number, then the sum a + b is irrational rational an integer

Explanation:

Step1: Recall rectangle area formula

The area formula of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l = 6\sqrt{10}$ and $w=3\sqrt{2}$.

Step2: Multiply the length and width

$A=(6\sqrt{10})\times(3\sqrt{2})=(6\times3)\times\sqrt{10\times2}= 18\sqrt{20}$. But we can simplify $\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}$, so $A = 18\times2\sqrt{5}=36\sqrt{5}$.

Step3: Recall the definition of irrational numbers

An irrational number is a number that cannot be written as a fraction $\frac{p}{q}$ where $p,q\in\mathbb{Z}$ and $q
eq0$. $\sqrt{2}$ cannot be written as a fraction, $\frac{3}{4}$ is a fraction, $3.125=\frac{3125}{1000}=\frac{25}{8}$ is a fraction, and $7.9=\frac{79}{10}$ is a fraction.

Step4: Recall the property of sum of rational and irrational numbers

Let $a$ be rational and $b$ be irrational. Assume $a + b=c$ is rational. Then $b=c - a$. Since the difference of two rational numbers $c$ and $a$ is rational, this contradicts the fact that $b$ is irrational. So $a + b$ is irrational.

Answer:

1. D. $36\sqrt{5}$
2. A. $\sqrt{2}$
3. A. irrational