The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is $\left(\frac{x}{100}\right) \%$. If the relative errors in measuring the mass and the diameter are $6.0 \%$ and $1.5 \%$ respectively, the value of $x$ is ____.

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Units & Measurement

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is

(\frac{x}{100}) %

. If the relative errors in measuring the mass and the diameter are

6.0%

and

1.5%

respectively, the value of

x

is ____.

Answer: 1050

Writing the density as

ρ = \frac{m}{\frac{4}{3} π ( d /2 ) ^{3}} = \frac{6 m}{π d ^{3}}

, the fractional error carries that of the mass once and that of the diameter three times:

\frac{Δ ρ}{ρ} = \frac{Δ m}{m} + 3 \frac{Δ d}{d}

. With

\frac{Δ m}{m} = 6.0%

and

\frac{Δ d}{d} = 1.5%

, this is

6.0 + 3 (1.5) = 10.5%

. Expressed as

\frac{x}{100} %

, the value is

x = 1050

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Integer answer