Question 1

A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surfaces will be

Accepted Answer

Concept & Geometry Integration: To find the ratio of their curved surfaces, we first need to express the dimensions of both shapes in terms of a single variable. For a hemisphere, its height is inherently equal to its base radius. Hemisphere CSA = 2\pi r^2 Cone CSA = \pi rl (where l = \sqrt{r^2 + h^2}) Step-by-Step Solution: 1. Establish Dimensions: - Equal bases mean they share the same radius, r. - The height of a hemisphere is equal to its radius (r). - Since their heights are equal, the height of the cone (h) is also r. (h = r) 2. Calculate Slant Height of Cone (l): - l = \sqrt{r^2 + h^2} - Substitute h = r: l = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} 3. Calculate Both Curved Surface Areas: - Hemisphere CSA = 2\pi r^2 - Cone CSA = \pi r (r\sqrt{2}) = \sqrt{2}\pi r^2 4. Find the Ratio: - \frac{Hemisphere CSA}{Cone CSA} = (2\pi r^2/\sqrt{2)\pi r^2} - Ratio = (2/\sqrt{2)} = \frac{\sqrt{2} × \sqrt{2}}{\sqrt{2}} = \sqrt{2} - Ratio = \sqrt{2} : 1 Exam Strategy & Shortcut: Recognize that a cone with equal height and radius forms an isosceles right triangle in cross-section. Its slant edge ratio is \sqrt{2}. The hemisphere's CSA multiplier is 2, and the cone's is \sqrt{2}. The ratio (2/\sqrt{2)} simplifies to \sqrt{2}:1 directly. Common Pitfall: A common misstep is failing to realize that the height of a hemisphere is strictly equal to its radius. Without this deduction, the problem appears unsolvable due to missing information. Final Answer: Therefore, the correct answer is \sqrt{2} : 1.

Question 2

If the radius of a sphere is increased by $10\%$, then the volume will be increased by

Accepted Answer

Concept & Successive Percentage Change: The volume of a sphere is given by: V = (4/3) \pi r^3 Since V \propto r^3, a percentage increase in the radius acts as three successive percentage increases on the volume. Step-by-Step Solution: • Given: • Increase in radius = 10% • Calculation (Method 1: Formula): 1. Let the initial radius be r=10 units. Initial Volume \propto 10^3 = 1000. 2. New radius r' = 10 + 10% of 10 = 11 units. 3. New Volume \propto 11^3 = 1331. 4. Increase in volume = 1331 - 1000 = 331. 5. Percentage increase = ((331/1000)) × 100 = 33.1%. Exam Strategy & Shortcut: Use the successive percentage change formula a + b + (ab/100) for the three factors of 10%: First two dimensions: 10 + 10 + (10 × 10/100) = 21%. Combine this result with the third dimension: 21 + 10 + (21 × 10/100) = 31 + 2.1 = 33.1%. Alternatively, memorize that 1.1^3 = 1.331, directly indicating a 33.1% increase. Common Pitfall: Adding the percentages directly (10% × 3 = 30%) is the most common mistake. This ignores the compounding effect of the volume expanding in three dimensions simultaneously. Final Answer: Therefore, the correct answer is 33.1%.

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