Question

You measure the side of a square as \(24.2\) centimeters with a possible error of \(0.25\) centimeter. Using these measurements, determine the interval containing the possible areas of the square.

Short answer

The interval that contains the possible areas of the square, considering the measurement error, ranges from \(573.1025\) square cm to \(598.0025\) square cm.

Step-by-step solution

Identity Possible Side Measurements

First, identify the possible ranges of the square's side based on the given error. The square's side could be from \(24.2 - 0.25 = 23.95\) cm to \(24.2 + 0.25 = 24.45\) cm.

Calculate Minimum Possible Area

Next, calculate the minimum possible area by squaring the minimum possible side length. Thus, the minimum possible area is \(23.95^2 = 573.1025\) square cm.

Calculate Maximum Possible Area

Similarly, calculate the maximum possible area by squaring the maximum possible side length. Thus, the maximum possible area is \(24.45^2 = 598.0025\) square cm.

Key concepts

Estimating Possible Error

Understanding the concept of estimating possible error is crucial when conducting measurements. In our daily lives, we rarely encounter instruments that provide perfectly accurate measurements. There is always a degree of uncertainty associated with any measurement. The 'possible error' indicates the degree to which the true value could differ from the measured value. For instance, if you measure the side of a square with a ruler that has graduations of 1 millimeter, the possible error in your measurement could be up to 0.5 millimeters, because the actual measurement could be slightly more or less than the reading.

When estimating possible error, it's important to consider the precision of the measuring instrument and the method used to make the measurement. In the exercise, the error margin given is \(\pm 0.25\) centimeters, which means our measurement could be off by as much as 0.25 centimeters on either side of the measured value. This range is crucial for any further calculations that depend on this measurement.

Interval Estimation

Interval estimation expands upon the concept of possible error by providing a range, or interval, within which the true value is likely to fall. This technique is particularly useful when dealing with uncertainties in measurements. In our exercise, once we know the possible error in the measurement of the square's side, we use this information to establish lower and upper limits for the actual value.

The interval of the square's side is calculated by subtracting and adding the possible error to the measured value. Hence, we end up with a minimum and maximum possible measurement, that is \(23.95\) cm and \(24.45\) cm respectively. Remember, the purpose of interval estimation is not to pinpoint the exact value but to define the boundaries within which the true measurement lies. This becomes particularly important for error propagation in further calculations, like determining the area of a square.

Area Calculation of a Square

The area of a square is found by squaring its side length. The formula is simple: \(\text{Area} = \text{side} \times \text{side}\). When considering error estimation, this area calculation can become more complex. This is because errors in the side measurement directly affect the area's accuracy. In our exercise, to find the possible areas, we need to account for the smallest and largest values the side could realistically be, due to the error margin.

By calculating the area using the minimum estimated side length, \(23.95\) cm, we find the smallest possible area. Conversely, by using the maximum side length, \(24.45\) cm, we determine the largest possible area. The result is an interval that represents the smallest and largest possible areas for the square: \[573.1025\, \text{cm}^2\] and \[598.0025\, \text{cm}^2\] respectively. Understanding this concept is vital for students who need to provide accurate measurements and calculations in various scientific and engineering disciplines.