A cylindrical disc made of lead has radius 24 cm and thickness 2 cm. There is a hole in the center of radius 2 cm. If the density of lead is 182 grams per cubic inch, what is the mass of the disc in kilograms rounded to the nearest tenth? (1 inch = 2.54 cm) (1 kilogram=1,000 grams)
Answer (2)
Calculate the volume of the cylindrical disc with a hole using the formula V = π ( r o u t er 2 − r inn er 2 ) h , resulting in approximately 3593.98 cm 3 .
Convert the volume from cubic centimeters to cubic inches by dividing by ( 2.54 ) 3 , obtaining approximately 219.32 in 3 .
Calculate the mass in grams by multiplying the volume in cubic inches by the density of lead (182 grams per cubic inch), which gives approximately 39915.92 grams .
Convert the mass from grams to kilograms by dividing by 1000, and round to the nearest tenth, resulting in a final mass of 39.9 kg.
Explanation
Problem Analysis First, let's analyze the given information. We have a cylindrical lead disc with a hole in the center. The disc's outer radius is 24 cm, its inner radius (hole) is 2 cm, and its thickness is 2 cm. The density of lead is 182 grams per cubic inch. We need to find the mass of the disc in kilograms, rounded to the nearest tenth. We also know that 1 inch = 2.54 cm and 1 kilogram = 1000 grams.
Calculate the Volume in Cubic Centimeters Next, we need to calculate the volume of the lead disc. The volume of a cylinder is given by V = π r 2 h , where r is the radius and h is the height (or thickness in this case). Since there's a hole in the center, we need to subtract the volume of the inner cylinder (the hole) from the volume of the outer cylinder. So, the volume of the disc is given by: V = π ( r o u t er 2 − r inn er 2 ) h where r o u t er = 24 cm, r inn er = 2 cm, and h = 2 cm. Plugging in the values, we get: V = π ( 2 4 2 − 2 2 ) × 2 = π ( 576 − 4 ) × 2 = π ( 572 ) × 2 = 1144 π ≈ 3593.98 cm 3
Convert Volume to Cubic Inches Now, we need to convert the volume from cubic centimeters to cubic inches. We know that 1 inch = 2.54 cm, so 1 cubic inch = ( 2.54 ) 3 cubic centimeters. Therefore, to convert the volume from cubic centimeters to cubic inches, we divide by ( 2.54 ) 3 :
V in = ( 2.54 ) 3 V = ( 2.54 ) 3 3593.98 = 16.387 3593.98 ≈ 219.32 in 3
Calculate Mass in Grams Now that we have the volume in cubic inches, we can calculate the mass of the disc in grams. We know the density of lead is 182 grams per cubic inch. So, the mass is given by: m g = density × V in = 182 × 219.32 ≈ 39915.92 grams
Convert Mass to Kilograms Finally, we need to convert the mass from grams to kilograms. We know that 1 kilogram = 1000 grams. So, to convert the mass from grams to kilograms, we divide by 1000: m k g = 1000 m g = 1000 39915.92 ≈ 39.91592 kg
Round to Nearest Tenth We need to round the mass to the nearest tenth of a kilogram. So, rounding 39.91592 kg to the nearest tenth gives us 39.9 kg.
Final Answer Therefore, the mass of the lead disc is approximately 39.9 kg.
Examples
Understanding how to calculate the mass of a cylindrical object with a hole is useful in various real-world scenarios. For example, when designing metal components for machinery, engineers need to accurately determine the mass of parts to ensure proper balance and functionality. Similarly, in construction, knowing the mass of materials like pipes or supports is crucial for structural integrity and safety. Even in jewelry making, calculating the mass of precious metals used in rings or bracelets helps determine their value and ensures the design meets specific weight requirements. These calculations rely on understanding volume, density, and unit conversions, just like in our lead disc problem.
The mass of the lead disc with a hole in the center is approximately 39.9 kg. This was calculated by determining the volume of the disc, converting it to cubic inches, and applying the density of lead. The final mass was rounded to the nearest tenth of a kilogram.
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