On a part-time job, you are asked to bring a cylindrical iron rod of length 85.8 cm and diameter 2.85 cm from a storage room to a machinist. Will you need a cart? (To answer, calculate the weight of the rod.)

검증된 단계별 안내

First, calculate the volume of the cylindrical rod using the formula for the volume of a cylinder: \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length) of the cylinder. Convert the diameter to radius by dividing by 2, and ensure all measurements are in meters.

Convert the length and diameter from centimeters to meters. The length \( h = 85.8 \text{ cm} = 0.858 \text{ m} \) and the diameter \( d = 2.85 \text{ cm} = 0.0285 \text{ m} \). Therefore, the radius \( r = \frac{0.0285}{2} \text{ m} \).

Substitute the values into the volume formula: \( V = \pi \times (0.01425)^2 \times 0.858 \). Calculate the volume in cubic meters.

Next, find the mass of the rod using the density of iron. The density of iron is approximately \( 7870 \text{ kg/m}^3 \). Use the formula \( m = \rho V \), where \( \rho \) is the density and \( V \) is the volume.

Finally, calculate the weight of the rod using the formula \( W = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)). Determine if the weight is manageable to carry or if a cart is needed.

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주요 개념

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

Volume of a Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height (or length) of the cylinder. This formula is essential for determining the amount of space the cylindrical rod occupies, which is a necessary step in calculating its weight.

Density

Density is defined as mass per unit volume, typically expressed in kilograms per cubic meter (kg/m³). For iron, a common density value is approximately 7,850 kg/m³. Knowing the density allows us to convert the volume of the rod into its mass, which is crucial for determining the weight.

Weight Calculation

Weight is the force exerted by gravity on an object, calculated as the product of mass and gravitational acceleration (W = mg). On Earth, the standard gravitational acceleration is approximately 9.81 m/s². By calculating the mass of the rod using its volume and density, we can determine its weight to decide if a cart is needed.