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Chapter 1: Problem 50

The outside dimensions of a cylindrical soda can are reported as \(12.559 \mathrm{~cm}\) for the diameter and \(5.62 \mathrm{~cm}\) for the height. (a) How many significant figures will the total outside area have: (1) two, (2) three, (3) four, or (4) five? Why? (b) What is the total outside surface area of the can in square centimeters?

Short Answer

Expert verified
(a) Three significant figures; (b) 469 cm虏.

Step by step solution

01

Identify the Number of Significant Figures

The given measurements are 12.559 cm (diameter) and 5.62 cm (height). The number of significant figures in the diameter is five, and the number of significant figures in the height is three. The total outside surface area is calculated using these dimensions, and the result should have significant figures equal to the measurement with the least number of significant figures. Therefore, the total outside area will have three significant figures.
02

Formula for Total Surface Area of a Cylinder

The formula to calculate the total surface area of a cylinder is given by:\[ A = 2\pi r (r + h) \]where \( r \) is the radius and \( h \) is the height of the cylinder.
03

Calculate the Radius

Since the diameter of the can is given as 12.559 cm, the radius \( r \) is:\[ r = \frac{12.559}{2} = 6.2795 \text{ cm} \]
04

Calculate the Total Surface Area

Now, substitute the values into the formula for the surface area:\[ A = 2\pi (6.2795) ((6.2795) + 5.62) \]\[ A \approx 2 \times 3.1416 \times 6.2795 \times 11.8995 \]Calculate the value inside the parentheses first:\[ 6.2795 + 5.62 = 11.8995 \]Now multiply:\[ 2 \times 3.1416 \times 6.2795 \times 11.8995 \approx 468.609 \text{ cm}^2 \]
05

Round to Correct Significant Figures

The calculated surface area should be rounded to the measurement with the least significant figures, which is three significant figures. Therefore, the total outside surface area is approximately:\[ 469 \text{ cm}^2 \] after rounding.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder Surface Area
Understanding the surface area of a cylinder is integral to this problem. A cylinder, like a soda can, is a three-dimensional shape with two circular bases and one curved surface connecting these bases. To find the total surface area, we use the formula:\[ A = 2\pi r (r + h) \]Where:- \( r \) is the radius of the circular base.- \( h \) is the height of the cylinder.This formula adds the areas of the two circles on each end and the rectangle wrapping around the side. The curved surface, when 'unwrapped,' forms a rectangle with a width of the circumference of the circle and a height of the cylinder. Remember, precision matters. Use precise measurements for accurate results. A tiny change in dimensions can affect the final answer significantly. Always consider both the measurements stated and how they influence the collected data's precision.
Measurement Precision
Measurement precision indicates the exactness and reliability of a given measurement. This plays a crucial role when working with significant figures. In the context of the soda can, diameter measures 12.559 cm, while height measures 5.62 cm. Each number has a different amount of significant figures: - The diameter, 12.559, has five significant figures. - The height, 5.62, has three significant figures. When performing calculations, the outcome's precision is determined by the measurement with the fewest significant figures. This ensures that the result does not falsely appear more precise than the least precise input. Therefore, the total surface area will have three significant figures due to the height measurement.
Diameter and Height Calculations
To accurately solve any problem involving cylinders, understanding how to work with diameter and height is essential. The diameter of the cylinder provides double the radius, the measure from the center to the edge of the circle, which is essential for calculations. The given diameter of 12.559 cm means a radius of:\[ r = \frac{12.559}{2} = 6.2795 \text{ cm} \]For calculations, working with the radius often simplifies the process. Hence, calculating radius correctly and precisely is crucial. As for the height (5.62 cm), it remains a straightforward linear dimension used directly in the formula.When inputs are plugged into the surface area formula:- Double-check each calculation step.- Use proper units.- Round off the final results using the input dimension having the least significant figures to maintain accuracy and consistency. Thus, it's all about taking the given dimensions, converting relevant measurements, and maintaining consistency throughout the calculations.

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