A copper refinery produces a copper ingot weighing 150 lb. If the copper is drawn into wire whose diameter is 7.50 mm, how many feet of copper can be obtained from the ingot? The density of copper is 8.94 g/cm³. (Assume that the wire is a cylinder whose volume 𝑉=𝜋𝑟²ℎ, where r is its radius and h is its height or length.)

검증된 단계별 안내

Convert the weight of the copper ingot from pounds to grams using the conversion factor: 1 lb = 453.592 g.

Calculate the volume of the copper ingot using the formula: \( \text{Volume} = \frac{\text{mass}}{\text{density}} \), where the mass is in grams and the density is given as 8.94 g/cm\(^3\).

Convert the diameter of the wire from millimeters to centimeters, and then calculate the radius by dividing the diameter by 2.

Use the formula for the volume of a cylinder \( V = \pi r^2 h \) to express the height (or length) \( h \) in terms of the volume and radius. Rearrange the formula to solve for \( h \): \( h = \frac{V}{\pi r^2} \).

Convert the length of the wire from centimeters to feet using the conversion factor: 1 cm = 0.0328084 ft.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Density

Density is defined as mass per unit volume and is a crucial property of materials. In this problem, the density of copper (8.94 g/cm³) allows us to convert the mass of the copper ingot into volume. Understanding how to manipulate the density formula (Density = Mass/Volume) is essential for determining how much space the copper occupies.

Volume of a Cylinder

The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height (or length) of the cylinder. In this context, the copper wire is modeled as a cylinder, and knowing this formula is necessary to relate the volume of the copper ingot to the length of wire that can be produced.

Unit Conversion

Unit conversion is the process of converting a quantity from one unit to another, which is often necessary in chemistry problems. In this case, converting the weight of the copper ingot from pounds to grams and the diameter of the wire from millimeters to centimeters is essential for consistent calculations. Mastery of unit conversions ensures accurate results in the final answer.