Problem Solving

If 0.0030<x−√30<0.0031, which of the following options is the approximation of (1/√30)−1/x?

A. 0.001
B. 0.002
C. 0.0001
D. 0.0002
E. 0.0003

The OA is C.

I need help to solve this PS question. Please, can any expert explain it for me? Thanks.

This is an interesting question.

First thing you have to notice is 0.0030<x-√30<0.0031 implies that $$x\approx\sqrt{30}.$$

By the other hand, $$\frac{1}{\sqrt{30}}-\frac{1}{x}=\frac{x-\sqrt{30}}{\sqrt{30}\cdot x}.$$

Now, as $$x\approx\sqrt{30},\ then\ \sqrt{30}\cdot x\approx\sqrt{30}\cdot\sqrt{30}=30.$$

Finally, if we divide 0.0030<x-√30<0.0031 by √30*x we will get $$\frac{0.0030}{\sqrt{30}\cdot x}<\frac{x-\sqrt{3}0}{\sqrt{30}\cdot x}<\frac{0.0031}{\sqrt{30}\cdot x}\ <=>\ \frac{0.003}{30}<\frac{x-\sqrt{30}}{\sqrt{30}\cdot x}<\frac{0.0031}{30}$$

After simplifiying we get $$0.0001<\frac{x-\sqrt{30}}{\sqrt{30}\cdot x}<0.000103.$$

So, the correct option is C.

I really hope this can help you.

I'm available if you'd like a follow up.