Approximation & Errors：練習問題

Round \(5.7382\) to 3 decimal places.

The length of a piece of string is measured as \(12.4 \text{ cm}\), correct to the nearest \(0.2 \text{ cm}\). Find the lower bound of the actual length.

A quantity A, representing an efficiency ratio, is calculated using the formula \(A = \frac{T^2}{V}\), where \(T\) is the time period and \(V\) is the volume.

The measurements are recorded as:
Time, \(T = 15.0 \text{ s}\) (corrected to the nearest \(0.1 \text{ s}\))
Volume, \(V = 2.0 \text{ m}^3\) (corrected to the nearest \(0.1 \text{ m}^3\))

Find the maximum possible percentage error in the calculated value of \(A\), relative to the value calculated using the recorded measurements. Give your answer correct to 3 significant figures.

The total surface area \(A\) of a closed cylindrical tank is calculated using the formula \(A = 2\pi r(r+h)\), where \(r\) is the radius and \(h\) is the height.

The measurements are recorded as:
Radius, \(r = 5.0\text{ cm}\) (corrected to 2 significant figures)
Height, \(h = 10\text{ cm}\) (corrected to 2 significant figures)

Using \(\pi = 3.142\), find the maximum possible percentage error in the calculated total surface area \(A\), relative to the value calculated using the recorded measurements. Give your answer correct to 3 significant figures.

The mass of a sample is measured as \(300 \text{ g}\), correct to the nearest \(10 \text{ g}\). The volume of the sample is measured as \(60 \text{ cm}^3\), correct to the nearest \(5 \text{ cm}^3\). Find the minimum possible density of the sample, giving your answer correct to 3 significant figures.

The length of a table is measured as \(1.2\) meters. If the actual length is exactly \(1.23\) meters, find the absolute error of the measurement.

A wire's length is measured as \(8.4\) cm, correct to the nearest \(0.1\) cm. Determine the maximum possible relative error of this measurement, expressing the answer as a fraction in its simplest form.

A decorator needs to buy enough paint to cover an area that requires \(2.3\) litres of paint. If paint is only sold in whole litre cans, how many cans of paint must be purchased?

A rectangular swimming pool is measured by a surveyor. The recorded dimensions, rounded to the nearest \(0.1 \text{ m}\), are:

• Length, \(L = 45.0 \text{ m}\)
• Width, \(W = 20.0 \text{ m}\)

(a) Determine the maximum absolute error in the measurement of the length, \(L\), and state the range of possible actual lengths using an inequality.

(b) Calculate the greatest possible area (Upper Bound) and the least possible area (Lower Bound) of the swimming pool, giving your answers in \(\text{m}^2\).

(c) Based on the measured dimensions, the calculated area is \(A = L \times W\). Find the maximum percentage error in the calculated area, correct to 3 significant figures.

(d) For preliminary rapid planning, the surveyor decides to estimate the area, \(A_{\text{est}}\), by rounding both the measured length and width to 1 significant figure before calculation. Calculate the percentage error of this estimation relative to the calculated area \(A\). Comment on whether this estimation strategy significantly increases the uncertainty compared to the inherent measurement error found in part (c).

A manufacturer measures the properties of a small metallic cube used in construction. The measurements are recorded as follows:

Mass: The mass, M, is measured as \(50.0 \text{ g}\), rounded to the nearest \(0.1 \text{ g}\).

Side Length: The side length, L, is measured as \(2.0 \text{ cm}\), rounded to the nearest \(0.1 \text{ cm}\).

The density \(\rho\) of the cube is calculated using the formula \(\rho = \frac{M}{L^3}\).

(a) Determine the maximum absolute error in the measurement of the mass M and the side length L.

(b) Determine the upper bound for the volume, \(V_{\text{max}}\), and the lower bound for the volume, \(V_{\text{min}}\), in \(\text{cm}^3\). Give your answers correct to 3 significant figures.

(c) Calculate the greatest possible density (Upper Bound, \(\rho_{\text{max}}\)) and the least possible density (Lower Bound, \(\rho_{\text{min}}\)) of the cube in \(\text{g/cm}^3\). Give your answers correct to 3 significant figures.

(d) Calculate the maximum percentage error in the calculated density \(\rho\), relative to the calculated value based on the recorded measurements. Give your answer correct to 2 decimal places.