Question

If x= 12 ​ − 9 ​ ,y= 13 ​ − 10 ​ , and z= 11 ​ − 8 ​ , then which of the following is true?

• A. z>x>y
• B. z>y>x
• C. y>x>z
• D. y>z>x

Answer 1

None of the provided options are correct.

Answer 2

Calculate the values:

$x = \sqrt{12} - \sqrt{9}, \quad y = \sqrt{13} - \sqrt{10}, \quad z = \sqrt{11} - \sqrt{8}$

To determine the relationship between x, y, and z, we can approximate their values or compare them analytically.

Let's analyze the function $f(a) = \sqrt{a+3} - \sqrt{a}$. We want to determine if this function is increasing or decreasing. $f'(a) = \frac{1}{2\sqrt{a+3}} - \frac{1}{2\sqrt{a}}$ Since $a+3 > a$, we have $\sqrt{a+3} > \sqrt{a}$, which means $\frac{1}{\sqrt{a+3}} < \frac{1}{\sqrt{a}}$. Thus, $f'(a) < 0$, so the function is decreasing.

Since 8 < 9 < 10, we have $f(8) > f(9) > f(10)$. Therefore, $\sqrt{11} - \sqrt{8} > \sqrt{12} - \sqrt{9} > \sqrt{13} - \sqrt{10}$, which means $z > x > y$.