Question

What is the value of ‘a’ when 1 micrometre is ${{10}^{-a}}$ hectometre?

Answer 1

We need to find the conversion value between the two given units of measurement. We can convert both of them into the standard unit and then compare to find the factor of ten by which the given scales of measuring differ from each other.

Complete answer:
We are given the two units of measuring the length. The micrometer and the hectometer are units of length. We can understand that from the ‘meter’ term in their names. We know that the standard unit of measuring length is the meters. So, what we need to do here is to convert the given two scales into the terms of meter and compare them to find the factor of ten by which they differ.
For converting each of the given scales of measurement to meters, we need to know their relation between the standard unit of measurement. We can go through the different units of measuring distance in the standard system.

& {{10}^{3}}m=1km \\\ & {{10}^{2}}m=1\text{hectometer} \\\ & {{10}^{-2}}m=1cm \\\ & {{10}^{-3}}m=1mm \\\ & {{10}^{-6}}m=1\mu m \\\ & {{10}^{-9}}m=1nm \\\ & {{10}^{-10}}m=1\overset{\text{0}}{\mathop{\text{A}}}\, \\\ \end{aligned}$$ The above given are the major units of measuring length in the standard system. We can utilise this data to get the answer we need. From this, we understand that, $${{10}^{2}}m=1\text{hectometer}$$ And, $${{10}^{-6}}m=1\mu m$$ Now, let us relate the two quantities by dividing the two relations as – $$\begin{aligned} & \dfrac{1\mu m}{1hm}=\dfrac{{{10}^{-6}}m}{{{10}^{2}}m} \\\ & \Rightarrow \text{ }1\mu m=\dfrac{{{10}^{-6}}m}{{{10}^{2}}m}hm \\\ & \therefore 1\mu m={{10}^{-8}}hm \\\ \end{aligned}$$ From the above calculation we understand that the micrometer and the hectometer is related to each other by a factor of ten as – $$\begin{aligned} & 1\mu m={{10}^{-8}}hm \\\ & \\\ \end{aligned}$$ The value of ‘a’ is the power of ten in this relation. $$\therefore a=8$$ **The value of ‘a’ is 8 from the above conversions. This is the required answer.** **Note:** The units of measuring length vary with the purpose or the quantity of the measurement. We use picometers and fermi to measure the dimensions of atoms and nuclei, whereas we use kilometers to measure the distance between two places. This is to avoid sophistication in data.