When measuring with a ruler, we may record a string's length as 8.2 cm. However, due to limitations of the ruler, the actual measurement could be 8.23 cm, making 8.2 cm the limit of accuracy with a precision of 0.1 cm. In this article, we will explore the concept of limits of accuracy and how it affects measurements in different scenarios. Let's start by defining what limits of accuracy are exactly.
Limits of accuracy refer to the possible range of values a measurement may have based on its rounded value. For instance, if a room's length is recorded as 10 meters with a tolerance of 1 meter, the accuracy is limited to 1 meter. Similarly, if someone's weight is rounded to the nearest 5 kg at 65 kg, the degree of accuracy is 5 kg.
In situations where a measurement is recorded with decimals, such as 10.2 cm rounded to the nearest 0.1 cm or £110 rounded to the nearest £10, the degree of accuracy is determined by the smallest unit of measure.
For example, if we agree to meet at 10 am, it could mean any time between 9:50 am and 10:10 am. This range, also known as the lower and upper limits, defines the degree of accuracy for the given time. Any time outside this range would be considered early or late.
To determine the limits of accuracy, we use a simple formula to calculate the upper and lower bounds. First, let's define what these terms mean.
The lower bound is the smallest possible value the measurement could be when rounded, while the upper bound is the largest possible value. The formula for the upper bound is the degree of accuracy divided by two plus the actual measurement, while the lower bound is determined by subtracting the degree of accuracy divided by two from the measurement.
Let's take a look at an example to better understand how to find upper and lower bounds.
If a bag of oranges is recorded as weighing 3 kg, rounded to the nearest kilogram, what could be the upper and lower bounds for the weight of the oranges?
The oranges could weigh 2.8 kg, 2.6 kg, or any weight less than 3 kg. To calculate the lower bound, we subtract half of the degree of accuracy, which is 1 kg, from 3 kg, giving us a lower bound of 2.5 kg. Calculating the upper bound is a bit more complicated. We know that 3 kg rounds up to 4 kg, but there is no single largest value that would round down to 3 kg. This can be seen with 3.5 kg, which rounds up to 4 kg. As a result, the upper bound is 3.5 kg, and the lower bound is 2.5 kg.
Let's consider a common question about limits of accuracy that one may come across in a GCSE exam.
If the average mass of an apple is recorded as 175 g, rounded to the nearest gram, what could be the smallest and largest possible masses for the apple?
The accuracy is limited to 1 gram in this case. To calculate the lower bound, we subtract half of the degree of accuracy, which is 0.5 g, from 175 g, giving us a lower bound of 174.5 g. Similarly, we add 0.5 g to 175 g to get an upper bound of 175.5 g. Using <, we can represent the range of possible values as an error interval, indicating that the mass of the apple cannot be exactly 175.5 g but can be any value less than that.
An error interval refers to the potential range of values a measurement can have, expressed through an inequality. This concept can be better understood through examples.
For instance, imagine a car traveling at a speed of 70 mph, rounded to the nearest 10 mph.
Solution: The degree of accuracy for this measurement is 10 mph. By dividing this value in half, we get 5 mph. Adding 5 mph to 70 mph, we get the upper bound of 75 mph. Subtracting 5 mph from 70 mph, we get the lower bound of 65 mph. If we define the speed of the car as x mph, the possible speeds fall within the range of x ≤ 75 and x ≥ 65.
Applying Limits of Accuracy: This concept is often used in GCSE-style questions, where the limits of accuracy play a key role in solving problems.
For instance, let's say Matthew measures the height and base of a triangle. He finds the height to be 16 cm rounded to the nearest centimeter and the base to be 20 cm rounded to the nearest 10 centimeters. What is the maximum possible area for the triangle?
Solution: We can use the formula for the area of a triangle: A = ½bh.
The maximum possible area can be achieved by using the highest possible values for both the base and height. The upper bound for the height is 16.5 cm, and the upper bound for the base is 25 cm. Thus, the maximum possible area is ½(25)(16.5) = 206.25 cm2.
Now, let's consider Tony who went on a 40-mile bike ride rounded to the nearest 10 miles. He traveled at a constant speed of 15 mph rounded to the nearest mph. What is the minimum time it took him to complete the bike ride?
Solution: Rearranging the formula for distance, d=rt, we get t=d/r. In order to minimize the time, we need to divide the smallest distance by the largest speed. The minimum distance is 35 miles (since 40 has been rounded to the nearest 10) and the maximum speed is 15.5 mph. Thus, the minimum time is 2.26 hours.
Next, let's imagine a triangle ABC with side AB equal to 10 cm and angle ACB equal to 34 degrees (rounded to the nearest degree).
Solution: We can use trigonometry to determine the upper and lower bounds for the length of AB. If you are unfamiliar with trigonometry, we recommend reviewing it before proceeding.
To find the lower and upper bounds, we must use the trigonometric ratio tan. Recall that tan = opposite/adjacent. In this case, we are trying to find the opposite side, so we can say that tan 34 = AB/10.
For the lower bound, we need to find the values of AB and 34 that result in the smallest possible value for the opposite side. This is achieved when AB = 9.5 cm and 34 is the lower bound for angle ACB. Thus, the lower bound for the opposite side is 5.503 cm.
Similarly, for the upper bound, we need to find the values of AB and 34 that result in the largest possible value. This is achieved when AB = 10.5 cm and 34 is the upper bound for angle ACB. Thus, the upper bound for the opposite side is 6.402 cm.
Therefore, the lower bound for side AB is 5.503 cm and the upper bound is 6.402 cm.
The limit of accuracy of a measurement refers to the range of possible values a measurement can be when rounded. The lower bound is the smallest possible value, while the upper bound is the largest possible value. Inequalities are used to define the error interval of a measurement.