Question 1

How many lead shots each $3$ mm in diameter can be made from a cuboid of dimensions $9 	ext{ cm} 	imes 11 	ext{ cm} 	imes 12 	ext{ cm}$?

Accepted Answer

Concept & Conservation of Volume and Unit Conversion: To find the number of identical small objects formed from melting a larger object, divide the volume of the larger object by the volume of a single small object. All measurements must be converted to the same unit before calculation. n = \frac{Volume of Cuboid}{Volume of one Lead Shot} Step-by-Step Solution: • Given: Cuboid dimensions = 9 cm × 11 cm × 12 cm. • Volume of Cuboid = 9 × 11 × 12 = 1188 cm^3. • Diameter of lead shot = 3 mm. Convert to cm: 3 mm = 0.3 cm. • Radius of lead shot r = (0.3/2) = 0.15 cm (or (3/20) cm). • Volume of one spherical shot = (4/3) × \pi × r^3. • = (4/3) × (22/7) × ((3/20))^3 = (4/3) × (22/7) × (27/8000). • = (88/21) × (27/8000) = (11/7) × (9/1000) = (99/7000) cm^3. • Number of shots n = (1188/\frac{99){7000}} = 1188 × (7000/99). • Divide 1188 by 99: 1188 \div 99 = 12. • n = 12 × 7000 = 84000. Exam Strategy & Shortcut: Work entirely in fractions and do not calculate intermediate decimal values. Set up the grand fraction: N = (9 × 11 × 12/\frac{4){3} × (22/7) × (3/20) × (3/20) × (3/20)}. Notice how the 11 in the numerator beautifully cancels the 22 in the denominator, reducing the math burden drastically. Common Pitfall: Failing to convert millimeters to centimeters. Mixing units will result in an answer off by a massive magnitude (like 72 or 84 instead of 84000). Final Answer: Therefore, the correct answer is 84000.

Question 2

In what ratio are the volumes of a cylinder, a cone and a sphere, if each has the same diameter and the same height?

Accepted Answer

Concept & Proportional Volumes: To compare the volumes of different geometric shapes with shared dimensions, express all dimensions in terms of a single variable, compute the standard formulas, and evaluate the final ratio. Step-by-Step Solution: 1. Identify Common Variables: Let diameter = d. Then radius r = (d/2) for all shapes. For a sphere, the height is its diameter. Thus, height h = d = 2r. Since all shapes have the same height, h = 2r for the cylinder and cone as well. 2. Express Volumes in terms of r: - Cylinder: V_{cyl} = \pi r^2h = \pi r^2(2r) = 2\pi r^3 - Cone: V_{cone} = (1/3)\pi r^2h = (1/3)\pi r^2(2r) = (2/3)\pi r^3 - Sphere: V_{sph} = (4/3)\pi r^3 3. Calculate the Ratio: V_{cyl} : V_{cone} : V_{sph} = 2\pi r^3 : (2/3)\pi r^3 : (4/3)\pi r^3 Divide by 2\pi r^3: = 1 : (1/3) : (2/3) Multiply by 3 to clear the fractions: = 3 : 1 : 2 Exam Strategy & Shortcut: If you recall Archimedes' principle regarding the cylinder and inscribed sphere, a sphere is (2/3) the volume of its circumscribed cylinder, and a cone is (1/3) of it. Thus the ratio of Cylinder:Cone:Sphere is 1 : (1/3) : (2/3), which scales immediately to 3 : 1 : 2. Common Pitfall: Misidentifying the "height" of the sphere. The height of a sphere is equal to its diameter (2r), not its radius (r). Final Answer: Therefore, the correct answer is 3 : 1 : 2.

Volume and Surface Area Questions

Questions

Practice smarter