Question

How to give my answer to an appropriate degree of accuracy?
\[R = \dfrac{{{x^2}}}{y}\]
\[x = 3.8 \times {10^5}\]
\[y = 5.9 \times {10^4}\]
What is the value of \[R\] giving an answer in standard form to an appropriate degree of accuracy?

Answer 1

Hint:
Here, we will substitute the values of the given variable in the given equation. We will then simplify it using the exponent rule. We will then use the basic mathematical operation to get the simplified value of \[R\]. Then we will write the answer to an appropriate degree of accuracy by observing the position of the decimal point in the question. The appropriate degree of accuracy is a measure of how close and correct a stated value is to the actual, real value being described.

Complete step by step solution:
It is given that \[R = \dfrac{{{x^2}}}{y}\], where, \[x = 3.8 \times {10^5}\] and \[y = 5.9 \times {10^4}\]
Substituting these values in \[R\], we get,
\[R = \dfrac{{{{\left( {3.8 \times {{10}^5}} \right)}^2}}}{{5.9 \times {{10}^4}}}\]
Using the identity \[{\left( {a \times b} \right)^m} = {a^m} \times {b^m}\] and \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\], we get,
\[ \Rightarrow R = \dfrac{{{{\left( {3.8} \right)}^2} \times {{10}^{5 \times 2}}}}{{5.9 \times {{10}^4}}} = \dfrac{{14.44 \times {{10}^{10}}}}{{5.9 \times {{10}^4}}}\]
Now, using the identity \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] and dividing \[14.44\] by \[5.9\], we get,
\[ \Rightarrow R = 2.447 \times {10^{10 - 4}} = 2.447 \times {10^6}\]
Now, we can see that the decimal values given in the question in \[x\] and \[y\] are to 1 decimal place, thus, we will give our answer to an appropriate degree of accuracy to 1 decimal place only.
Thus, we get,
\[R = 2.447 \times {10^6} \approx 2.4 \times {10^6}\]

Hence, the value of \[R\] giving an answer in standard form to an appropriate degree of accuracy is \[2.4 \times {10^6}\].
Thus, this is the required answer.

Note:
Accuracy may be affected by rounding, the use of significant figures or ranges in measurement. In maths “to an appropriate degree of accuracy” means that the question wants us to present our answer in the same form as the least accurate measure in the question. Also, we should know that the accuracy of a measurement or approximation is the degree of closeness to the exact value whereas the error is the difference between the approximation and the exact value. Hence, approximation and error are complete different terms.