Question Type 5: Working with errors and estimation Exercises

Two forces are measured as $A = 50\text{ N}$ and $B = 30\text{ N}$, each correct to the nearest $1\text{ N}$. The reported resultant difference is $R_0 = A - B = 20\text{ N}$. Find the maximum possible percentage error in $R_0$.

The radius of a sphere is measured as $3.1\text{ cm}$ correct to the nearest $0.1\text{ cm}$. Without using differentials, find the maximum possible percentage error in the surface area computed from $A=4\pi r^2$ using $r=3.1\text{ cm}$.

A car travels a reported distance of $125 \text{ km}$ in a reported time of $2.0 \text{ h}$, where distance is to the nearest $1 \text{ km}$ and time to the nearest $0.1 \text{ h}$. Find the maximum possible average speed in $\text{km/h}$ consistent with these measurements.

A ramp's rise is measured as $3.2\text{ m}$ and run as $8.0\text{ m}$, each correct to the nearest $0.1\text{ m}$. The reported gradient is $m_0=\frac{3.2}{8.0}=0.4$. Determine the maximum possible percentage error in $m_0$.

The radius of a sphere is reported as $12\text{ m}$ correct to the nearest metre. Without differentials, find the maximum possible percentage error in the volume computed from $V=\tfrac{4}{3}\pi r^3$ using $r=12\text{ m}$.

A rectangle has length $8.2\text{ cm}$ and width $5.7\text{ cm}$, each measured correct to the nearest $0.1\text{ cm}$. The reported area is $A_0=8.2\times 5.7=46.74\text{ cm}^2$. Find the maximum possible percentage error in $A_0$ relative to the true area.

A length is measured as $12.3\text{ cm}$ correct to the nearest $0.1\text{ cm}$. State the maximum absolute error and the maximum percentage error in the measurement.

A circular lid has measured diameter $d=2.4\text{ cm}$ correct to the nearest $0.1\text{ cm}$. Using $C=\pi d$, find the maximum possible percentage error in the circumference computed from $d=2.4\text{ cm}$.

A square’s side is given as $0.34 \text{ m}$ correct to $2$ significant figures. Find the maximum possible percentage error in the computed area $A=\text{side}^2$ using $0.34 \text{ m}$.

A triangle’s base and height are measured as $b=12.5\text{ cm}$ and $h=7.6\text{ cm}$, each to the nearest $0.1\text{ cm}$. The reported area is $A_0=\frac{1}{2} bh=47.5\text{ cm}^2$. Find the maximum possible percentage error in $A_0$.

An annulus has outer radius $R = 5.0\text{ cm}$ and inner radius $r = 2.0\text{ cm}$, each measured to the nearest $0.1\text{ cm}$. The reported area is $A_0 = \pi(R^2 - r^2) = 21\pi\text{ cm}^2$. Find the maximum possible percentage error in $A_0$.

A current is measured as $I = 2.50\text{ A}$ correct to the nearest $0.01\text{ A}$. Power $P$ in a resistor is proportional to $I^2$. Find the maximum possible percentage error in the computed power due solely to the measurement error in $I$.

A cylinder has measured radius $r = 4.0\text{ cm}$ and height $h = 10.0\text{ cm}$, each correct to the nearest $0.1\text{ cm}$. Find the upper and lower bounds for the volume $V = \pi r^2 h$.