Question

The height of a computer screen is $31.25\mathrm{cm}$, and its width is $47\mathrm{cm}$. (a) Is the area of the screen known to one, two, three, or four significant figures? (b) Calculate the area of the screen, giving your answer with the correct number of significant figures.

Short answer

(a) The area is known to two significant figures. (b) The area is 1500 cm².

Step-by-step solution

Identify Significant Figures in Given Measurements

In the problem, the height of the screen is given as $31.25\text{cm}$ which has four significant figures, and the width is given as $47\text{cm}$ which has two significant figures.

Determine the Significant Figures for the Area Calculation

When calculating with measurements, the result should be reported with the same number of significant figures as the measurement with the fewest significant figures. In this case, the width $47$ has the fewest with two significant figures.

Calculate the Area of the Screen

To find the area of the screen, use the formula for the area of a rectangle: $$\text{Area}=\text{height}\times \text{width}=31.25\text{cm}\times 47\text{cm}$$ This results in an area of $1468.75{\text{cm}}^2$.

Round the Area to the Correct Number of Significant Figures

Since the width measurement limits the significant figures to two, round $1468.75$ to two significant figures, resulting in $1500{\text{cm}}^2$.

Key concepts

Measurement Accuracy

Measurement accuracy refers to how close a measured value is to the true value. It's essential in science and everyday tasks because it ensures the reliability of results. A key part of understanding measurement accuracy is how it's often influenced by significant figures. Significant figures indicate the precision of a measurement. They provide insight into the certainty of a quantity. In our example, the height of the screen is reported as $31.25\text{cm}$, meaning it is precise up to four significant figures. Meanwhile, the width $47\text{cm}$ only has two significant figures, implying less precision.When dealing with measurements, always consider which measurement has the least precision (fewest significant figures), as it limits the accuracy of any calculated results. This principle ensures that we do not overstate the accuracy of a computed result.

Calculating with Significant Figures

Calculations often involve multiplying or dividing measured values. It's crucial to apply the correct rules of significant figures to these calculations to maintain the integrity of the results.Significant figures guide us in understanding how precise our final answer can be. The rule of thumb is that the result should have no more significant figures than the measurement with the fewest significant figures. In our scenario, the width $47\text{cm}$ has only two significant figures, thus constraining the final result's precision. When you calculate the area of the screen using the formula $\text{Area}=\text{height}\times \text{width}=31.25\text{cm}\times 47\text{cm}$, you initially get $1468.75{\text{cm}}^2$ as the unrounded result. To respect significant figure rules, round it to two significant figures, resulting in $1500{\text{cm}}^2$. This process prevents falsely assumed precision.

Area of Rectangles

The area of a rectangle is a simple calculation that involves multiplying its length by its width. It's a fundamental concept that is widely used in various fields such as architecture, engineering, and day-to-day activities like interior design.Understanding the area helps you determine how much space an object covers. For the computer screen in our example, the height is $31.25\text{cm}$ and the width is $47\text{cm}$. Thus, the formula for the area, $\text{Area}=\text{height}\times \text{width}$, is applied.The computation yields an area of $1468.75{\text{cm}}^2$, but due to significant figure considerations, the precise answer is reported as $1500{\text{cm}}^2$. Recognizing how dimensions translate into space coverage and applying significant figures properly ensures that your calculated areas are both useful and accurately presented.