Problem 108 - General Chemistry

Question

What is the difference in surface area between two circles, one of radius \(7.98 \mathrm{~cm},\) the other of radius \(8.50 \mathrm{~cm} ?\) The surface area of a circle of radius \(r\) is \(\pi r^{2}\). Obtain the result to the correct number of significant figures.

Step-by-Step Solution

Answer

The difference in surface area is approximately 26.9 cm².

Step 1: Write the formula for the area of the circle

The formula for the area of a circle is given by: \( A = \pi r^2 \), where \( A \) is the area and \( r \) is the radius.

Step 2: Calculate the area of the first circle

For the first circle with radius \( r_1 = 7.98 \) cm, substitute into the formula to get: \( A_1 = \pi \times (7.98)^2 \). Evaluating this gives \( A_1 \approx 200.039 \) cm².

Step 3: Calculate the area of the second circle

For the second circle with radius \( r_2 = 8.50 \) cm, substitute into the formula to get: \( A_2 = \pi \times (8.50)^2 \). Evaluating this gives \( A_2 \approx 226.981 \) cm².

Step 4: Calculate the difference in areas

The difference in the areas of the two circles is given by: \( \Delta A = A_2 - A_1 \). Substituting the values calculated gives \( \Delta A = 226.981 - 200.039 = 26.942 \) cm².

Step 5: Provide the result with correct significant figures

The original values have three significant figures (7.98 and 8.50). Therefore, the final result should also have three significant figures. Thus, \( \Delta A \) is approximately \( 26.9 \) cm².

Key Concepts

Understanding Significant Figures

Significant figures are critical when it comes to precision in mathematics and science. They determine how many digits should be used in a measurement or calculation to convey its accuracy. When measuring or calculating with significant figures, the digits that are known accurately plus one estimated digit are considered. For instance, the radii given in the exercise, 7.98 cm and 8.50 cm, both have three significant figures.

• "7.98 cm" has: 7, 9, and 8 as significant.
• "8.50 cm" has: 8, 5, and the trailing 0 as significant.

When performing calculations with these values, it's crucial to express your final result in three significant figures to ensure consistency with the precision of the measurements you started with. This is exactly why the solution rounded the area difference to 26.9 cm².

Surface Area Calculation of a Circle

Calculating the surface area of a circle is straightforward thanks to the formula: \[ A = \pi r^2 \]This means you multiply the square of the radius, \( r^2 \), by the constant \( \pi \) (approximately 3.14159). So, if you know the radius, you can find the area. In our exercise, we have two radii: 7.98 cm and 8.50 cm.

• For a radius of 7.98 cm, the area calculation becomes:\[ A_1 = \pi \times (7.98)^2 \approx 200.039 \, \text{cm}^2 \]
• For a radius of 8.50 cm, the area is:\[ A_2 = \pi \times (8.50)^2 \approx 226.981 \, \text{cm}^2 \]

This step is crucial because it allows us to find the area corresponding to each radius, paving the way for a comparison.

Calculating the Area of a Circle and Difference

The area of a circle is a fundamental concept in geometry, often used to quantify the space enclosed by a circle's perimeter. It allows us to understand how much 'flat' space the circle covers. To find this area, we use the formula \( A = \pi r^2 \), where \( r \) is the radius.

• In the exercise, the first circle with a radius of 7.98 cm gives an area of approximately 200.039 cm².
• The second circle with a radius of 8.50 cm results in an area of approximately 226.981 cm².

The difference in area between these two circles can be calculated by subtracting the smaller area from the larger one: \[ \Delta A = A_2 - A_1 \approx 226.981 - 200.039 = 26.942 \, \text{cm}^2 \]Finally, given the significant figures rules, we report the area difference as 26.9 cm², highlighting the minor differences between the two circles.