This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the mensuration problems. 9. The curved surface area of a cylinder of radius 7cm and height 10cm is: Step 1: Recall the formula for the curved surface area (CSA) of a cylinder. CSA = 2 rh where r is the radius and h is the height. Step 2: Substitute the given values into the formula. Given r = 7 cm and h = 10 cm. CSA = 2 × × 7 cm × 10 cm Step 3: Calculate the area. CSA = 140 cm^2 The correct option is A. The curved surface area is 140 cm^2. 10. The total surface area of a closed cylinder of radius 5cm and height 5cm is: Step 1: Recall the formula for the total surface area (TSA) of a closed cylinder. TSA = 2 r(r+h) where r is the radius and h is the height. Step 2: Substitute the given values into the formula. Given r = 5 cm and h = 5 cm. TSA = 2 × × 5 cm × (5 cm + 5 cm) TSA = 10 cm × (10 cm) Step 3: Calculate the area. TSA = 100 cm^2 The correct option is B. The total surface area is 100 cm^2. 11. The curved surface area of a right circular cone of radius 6cm and slant height 9cm is: Step 1: Recall the formula for the curved surface area (CSA) of a cone. CSA = rl where r is the radius and l is the slant height. Step 2: Substitute the given values into the formula. Given r = 6 cm and l = 9 cm. CSA = × 6 cm × 9 cm Step 3: Calculate the area. CSA = 54 cm^2 The correct option is C. The curved surface area is 54 cm^2. 12. The Volume of a cone is 154cm^3. If its height is 6cm and = (22)/(7), find its radius: Step 1: Recall the formula for the volume of a cone. V = (1)/(3) r^2 h where V is the volume, r is the radius, and h is the height. Step 2: Substitute the given values into the formula. Given V = 154 cm^3, h = 6 cm, and = (22)/(7). 154 cm^3 = (1)/(3) × (22)/(7) × r^2 × 6 cm Step 3: Simplify the equation. 154 = (22 × 6)/(3 × 7) r^2 154 = (132)/(21) r^2 154 = (44)/(7) r^2 Step 4: Solve for r^2. r^2 = 154 × (7)/(44) r^2 = (1078)/(44) r^2 = 24.5 Wait, let me recheck the calculation. r^2 = 154 × (7)/(44) = (11 × 14) × (7)/((11 × 4)) = (14 × 7)/(4) = (98)/(4) = 24.5 This value does not lead to a simple integer or half-integer radius from the options. Let me re-evaluate the simplification. 154 = (1)/(3) × (22)/(7) × r^2 × 6 154 = (22 × 6)/(21) r^2 154 = (132)/(21) r^2 154 = (44)/(7) r^2 r^2 = 154 × (7)/(44) Since 154 = 44 × 3.5, r^2 = (44 × 3.5) × (7)/(44) r^2 = 3.5 × 7 r^2 = 24.5 This is still 24.5. Let me check the options again. A. 3.5cm B. 5cm C. 9cm D. 14cm If r = 3.5, then r^2 = (3.5)^2 = 12.25. If r = 5, then r^2 = 25. It seems there might be a slight discrepancy in the problem or options, as 24.5 is not a perfect square that matches the options. However, 24.5 is very close to 25. Let's assume the closest option is intended. If r^2 = 25, then r = 5 cm. Let's check if r=5 cm gives V=154 cm^3. V = (1)/(3) × (22)/(7) × 5^2 × 6 = (1)/(3) × (22)/(7) × 25 × 6 = (22 × 25 × 2)/(7) = (1100)/(7) ≈ 157.14 cm^3. This is not exactly 154 cm^3. Let's re-examine the calculation 154 × (7)/(44). 154 / 44 = 3.5. 3.5 × 7 = 24.5. So r^2 = 24.5. If the volume was 154 × (25)/(24.5) ≈ 157.14, then r would be 5. If the volume was 154, then r = sqrt(24.5) ≈ 4.95. The closest option to 4.95 is 5 cm. It's possible the problem intends for r=5 and the volume is an approximation, or there's a typo in the volume or height. Given the multiple-choice format, we should select the closest reasonable answer. Let's assume the question intended for r=3.5 cm. If r = 3.5 cm, then r^2 = (3.5)^2 = 12.25. V = (1)/(3) × (22)/(7) × (3.5)^2 × 6 = (1)/(3) × (22)/(7) × 12.25 × 6 V = (22 × 12.25 × 2)/(7) = (22 × 24.5)/(7) = 22 × 3.5 = 77 cm^3. This is not 154 cm^3. Let's assume the question intended for r=7 cm. If r = 7 cm, then r^2 = 49. V = (1)/(3) × (22)/(7) × 7^2 × 6 = (1)/(3) × (22)/(7) × 49 × 6 = 22 × 7 × 2 = 308 cm^3. This is not 154 cm^3. Let's re-check the calculation for r^2 = 24.5. If r^2 = 24.5, then r = sqrt(24.5) ≈ 4.9497. The closest option is 5 cm. Let's consider if the height was different. If r=5 and V=154, then 154 = (1)/(3) × (22)/(7) × 5^2 × h. 154 = (22 × 25)/(21) h. h = 154 × (21)/(550) = (3234)/(550) ≈ 5.88 cm. This is close to 6 cm. Given the options, and the common practice in such problems, it's highly probable that r=5 cm is the intended answer, with a slight rounding in the volume or height. Let's proceed with r=5 cm as the most plausible answer among the choices. The value of r is approximately 4.95 cm, which is closest to 5 cm. The correct option is B. The radius is 5 cm. 13. A closed cylindrical tank has radius 7cm and height 12cm. Calculate: i) Its total surface area Step 1: Recall the formula for the total surface area (TSA) of a closed cylinder. TSA = 2 r(r+h) where r is the radius and h is the height. Step 2: Substitute the given values into the formula. Given r = 7 cm, h = 12 cm, and = (22)/(7). TSA = 2 × (22)/(7) × 7 cm × (7 cm + 12 cm) TSA = 2 × 22 × (19) cm^2 Step 3: Calculate the area. TSA = 44 × 19 cm^2 TSA = 836 cm^2 The total surface area is 836 cm^2. ii) Its volume (Take = (22)/(7)) Step 1: Recall the formula for the volume of a cylinder. V = r^2 h where r is the radius and h is the height. Step 2: Substitute the given values into the formula. Given r = 7 cm, h = 12 cm, and = (22)/(7). V = (22)/(7) × (7 cm)^2 × 12 cm V = (22)/(7) × 49 cm^2 × 12 cm Step 3: Simplify and calculate the volume. V = 22 × 7 × 12 cm^3 V = 154 × 12 cm^3 V = 1848 cm^3 The volume is 1848 cm^3. 14. A Cone has radius 5cm and slant height 13cm. i) Calculate its Curved surface area. Step 1: Recall the formula for the curved surface area (CSA) of a cone. CSA = rl where r is the radius and l is the slant height. Step 2: Substitute the given values into the formula. Given r = 5 cm and l = 13 cm. CSA = × 5 cm × 13 cm Step 3: Calculate the area. CSA = 65 cm^2 The curved surface area is 65 cm^2. ii) If the vertical height of the Cone is 12cm, Calculate its volume (Take = (22)/(7)) Step 1: Recall the formula for the volume of a cone. V = (1)/(3) r^2 h where r is the radius and h is the vertical height. Step 2: Substitute the given values into the formula. Given r = 5 cm, h = 12 cm, and = (22)/(7). V = (1)/(3) × (22)/(7) × (5 cm)^2 × 12 cm V = (1)/(3) × (22)/(7) × 25 cm^2 × 12 cm Step 3: Simplify and calculate the volume. V = (22 × 25 × 12)/(3 × 7) cm^3 V = (22 × 25 × 4)/(7) cm^3 V = (2200)/(7) cm^3 V ≈ 314.29 cm^3 The volume is (2200)/(7) cm^3. 3 done, 2 left today. You're making progress.