ઘન ધાતુના ગોળાની ઘનતા તેનું દળ અને વ્યાસ માપીને કરવામાં આવે છે. ધાતુના ગોળાની ઘનતામાં મહત્તમ ત્રુટિ \(\left(\frac{x}{100}\right) \% \) છે. જો દળ અને વ્યાસના માપનમા સાપેક્ષ ત્રુટિ અનુક્રમે \(6.0 \%\) અને \(1.5 \%\) હોય તો \(x\) નું મૂલ્ય કેટલુ હશે?

A \(1000\)
B \(1075\)
C \(1060\)
D \(1050\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(1050\)

Step-by-step Solution

Detailed explanation

\(\rho=\frac{M}{V}=\frac{M}{\frac{4}{3} \pi\left(\frac{D}{2}\right)^{3}}\) \(\rho=\frac{6}{\pi} M D ^{-3}\) taking log \(\ell n \rho=\ell n \left(\frac{6}{\pi}\right)+\ell n M -3 \ell m D\) Differentiates \(\frac{ d p}{\rho}=0+\frac{ d M }{ M }-3 \frac{ d ( D )}{ D }\) for…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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