Geometry and Mensuration

The area of a trapezium is calculated using the formula: Area = 1/2 × (sum of parallel sides) × height. Here, (20 + 18) = 38 cm, and height = 15 cm, so area = 1/2 × 38 × 15 = 285 cm². However, the calculation shows 285 cm², which corresponds to option C, so the original correct answer C is correct. The explanation was missing and has been added. Difficulty is easy given the straightforward calculation.

The area of a triangle can be found using the formula Area = inradius × semiperimeter. Given the perimeter is 28 cm, the semiperimeter is 14 cm. Multiplying by the inradius 2.5 cm gives an area of 35 cm². However, since 35 cm² is not listed, the closest correct calculation is 2.5 × 14 = 35 cm², so the correct answer should be 35 cm². Since 35 cm² is not an option, the options are incorrect. The best approach is to correct the options to include 35 cm² and select it as correct. Alternatively, if options cannot be changed, then 'None of these' would be correct. But here, options do not include 35 cm², so the correct answer is 'None of these'.

First, find the radii from the circumferences: r = C/(2π). For the smaller circle, r = 264/(2π) = 42 meters; for the larger, r = 352/(2π) = 56 meters. Then, calculate areas: A = πr². Smaller area = π×42² = 1764π; larger area = π×56² = 3136π. The difference is 3136π - 1764π = 1372π ≈ 4304 sq m.

The side length of the square plot is the square root of 289, which is 17 ft. The perimeter is 4 times the side, so 4 × 17 = 68 ft. The cost of building the fence is 68 ft × Rs. 58 = Rs. 3944, which corresponds to option A.

The volume of the metallic sphere is conserved when it is melted and drawn into a wire. The volume of the sphere is given by $\frac{4}{3}\pi r^3=\frac{4}{3}\pi (12{)}^3=7238.23{\text{ cm}}^3$. The wire is a cylinder with volume $\pi r^{2}h$, where the radius of the wire's cross section is 1.6 cm and $h$ is the length. Equating the volumes: $7238.23=\pi (1.6{)}^{2}h$, solving for $h$ gives $h=\frac{7238.23}{\pi \times 2.56}\approx 900\text{ cm}$ or 9 meters. Since none of the given options (45 cm, 18 cm, 90 cm) are close to 900 cm, the correct choice is "None of these" (option D).

The area of a rhombus is given by the formula: Area = (1/2) × (product of diagonals). Given the side length is 25 cm and one diagonal is 30 cm, we use the Pythagorean theorem to find the other diagonal. Half of the known diagonal is 15 cm. Using the right triangle formed by half the diagonals and the side: (other diagonal/2)^2 + 15^2 = 25^2, so (other diagonal/2)^2 = 625 - 225 = 400, thus other diagonal = 40 cm. The area is then 0.5 × 30 × 40 = 600 cm². Therefore, the correct answer is option C (600 sq.cm). The units in the options should be cm², not sq.m as originally stated.

Let the sides of the rectangle be x and y. The perimeter is 2(x + y) = 28, so x + y = 14. The area of the rectangle is xy. The square formed on the diagonal has side equal to the diagonal length, which is $\sqrt{x^2+y^2}$, so its area is x^2 + y^2. The problem states this area is 108 1/3 % (which is 13/12 times) more than the area of the rectangle, so: x^2 + y^2 = xy + (13/12)xy = (25/12)xy. Using (x + y)^2 = x^2 + y^2 + 2xy, we get 14^2 = (25/12)xy + 2xy = (49/12)xy, so xy = (14^2 * 12)/49 = 48. Then, (x - y)^2 = (x + y)^2 - 4xy = 196 - 192 = 4, so x - y = 2. Hence, the difference between the sides is 2 units.

The wire originally forms a circle with radius 3.5 m, so its length is the circumference of the circle: 2πr = 2 × π × 3.5 ≈ 22 m. When bent into a rectangle with length and breadth in the ratio 6:5, let length = 6x and breadth = 5x. The perimeter of the rectangle is 2(6x + 5x) = 22x, which equals the wire length 22 m. Solving 22x = 22 gives x = 1. Therefore, length = 6 m and breadth = 5 m. The area of the rectangle is length × breadth = 6 × 5 = 30 m². The options given are in cm² but with very small values, which is inconsistent with the problem's scale. The correct area is 30 m², matching option B if units are meters squared. Hence, the correct answer is B.

The problem provides the area of the rectangle and a relation between the perimeter of the square and the rectangle, but without specific dimensions or additional constraints, the area of the square cannot be uniquely determined.

The side of the square is half the radius of the larger circle, so the side length is √196 = 14 cm. Therefore, the radius of the larger circle is 2 × 14 = 28 cm. The radius of the smaller circle is (3/7) × 28 = 12 cm. The circumference of the smaller circle is 2π × 12 = 24π cm.

The side of the square is the radius of the circle. Since the area of the square is 1225 sq.units, the side length (radius) is √1225 = 35 units. The length of the rectangle is two-fifths of the radius, so length = (2/5) × 35 = 14 units. Given the breadth is 10 units, the area of the rectangle = length × breadth = 14 × 10 = 140 sq.units. However, the options show 140 as A, but the correct calculation gives 140, so the correct answer is A, not C. Rechecking calculations: length = (2/5)*35=14, area=14*10=140. So the original correct_answer 'A' is correct. Therefore, the correct answer is A.

Given the curved surface area (CSA) of the cylinder is 264 m² and volume is 924 m³, we use the formulas: CSA = 2πrh and Volume = πr²h. From CSA, 2πrh = 264, so rh = 264/(2π). From volume, πr²h = 924, so r²h = 924/π. Dividing volume by CSA: (πr²h) / (2πrh) = r/2 = 924/264 = 3.5, so r = 7. Using rh = 264/(2π) and substituting r=7, h = 6. Diameter = 2r = 14. Therefore, the ratio of diameter to height = 14:6 = 7:3. This matches option B. The initial explanation incorrectly matched the ratio to option D, but the correct ratio is 7:3, option B.

The surface area of a sphere is given by the formula 4πr². If the radius is doubled, the new radius becomes 2r. Substituting this into the formula gives the new surface area as 4π(2r)² = 4π × 4r² = 4 × (4πr²), which is four times the original surface area. Given the original surface area is 2464 cm², the new surface area will be 4 × 2464 = 9856 cm². Therefore, the correct answer is option C (9856 cm²).

The external dimensions of the cistern are 3.3 m by 2.6 m by 1.1 m, with walls 5 cm thick. Converting to centimeters: length = 330 cm, width = 260 cm, height = 110 cm. The walls reduce the internal length and width by twice the wall thickness (5 cm on each side), so internal length = 330 - 10 = 320 cm, internal width = 260 - 10 = 250 cm. The internal volume is given as 8000 liters, which is 8,000,000 cubic centimeters. Using the volume formula: 320 × 250 × internal height = 8,000,000, solving gives internal height = 100 cm. The external height is 110 cm, so the thickness of the bottom is 110 - 100 = 10 cm, which is 1 dm. Therefore, the correct answer is option B (1 dm).

The total surface area of a right circular cone is given by πr(l + r), where r is the radius and l is the slant height. Here, r = 7 cm and height h = 14 cm. The slant height l = √(r² + h²) = √(49 + 196) = √245 ≈ 15.65 cm. Therefore, total surface area = π × 7 × (15.65 + 7) ≈ 3.1416 × 7 × 22.65 ≈ 497.42 cm², which does not match any of the given options A, B, or C. Hence, the correct answer is D (None of these).

The volume of earth dug from the rectangular field equals the volume of the platform formed, adjusted by the increase factor (10/5 = 2). Calculating the platform volume and dividing by the field area gives the required depth. The correct depth is 12 cm, not 10 cm.

The curved surface area (CSA) of a cylinder is given by the formula CSA = 2πrh, where r is the radius and h is the height. Given the radius as 2r and height as 3r, substituting these values gives CSA = 2 × π × 2r × 3r = 12πr². Therefore, the correct curved surface area is 12πr², which corresponds to option C.

The volume of the original sphere is $\frac{4}{3}\pi (10{)}^3=\frac{4000}{3}\pi$. Dividing this volume equally into 8 smaller spheres, each smaller sphere has volume $\frac{4000}{3}\pi ÷8=\frac{500}{3}\pi$. Using the volume formula for a sphere, $\frac{4}{3}\pi r^3=\frac{500}{3}\pi$, solving for $r$ gives $r=5$. The surface area of each smaller sphere is $4\pi r^2=4\pi (5{)}^2=100\pi$.

The volume of soil extracted is calculated by multiplying the length, width, and depth of the pond: 20 m × 10 m × 5 m = 1000 cubic meters.

The volume of a cone is given by the formula (1/3) × base area × height. Given the base area is 30 cm² and the height is 6 cm, the volume = (1/3) × 30 × 6 = (1/3) × 180 = 60 cm³. Therefore, the correct answer is D, 60 cm³.

Let the radius of the base be r. The volume of the cone is $\frac{1}{3}\pi r^{2}h$. Let the height of the cylinder be H. The volume of the cylinder is $\pi r^{2}H$. The total volume of the solid is the sum of the volumes of the cylinder and the cone: $\pi r^{2}H+\frac{1}{3}\pi r^{2}h$. According to the problem, this total volume is three times the volume of the cone, so: $$\pi r^{2}H+\frac{1}{3}\pi r^{2}h=3\times \frac{1}{3}\pi r^{2}h=\pi r^{2}h$$ Dividing both sides by $\pi r^2$ gives: $$H+\frac{1}{3}h=h$$ Solving for H: $$H=h-\frac{1}{3}h=\frac{2}{3}h$$ Therefore, the height of the cylinder is $\frac{2}{3}h$, which corresponds to option C.

Given the ratio 6:5:4, let the dimensions be 6x, 5x, and 4x. The total surface area of a cuboid is 2(lb + bh + hl). Substituting, 2(6x*5x + 5x*4x + 6x*4x) = 33300, which simplifies to 2(30x^2 + 20x^2 + 24x^2) = 33300, or 2(74x^2) = 33300, so 148x^2 = 33300, giving x^2 = 225, and x = 15. Therefore, length = 6*15 = 90 cm, breadth = 5*15 = 75 cm, height = 4*15 = 60 cm, matching option D.

The curved surface area of a sphere is given by 4πr². Given 64π = 4πr², dividing both sides by 4π gives r² = 16, so r = 4 cm. However, the options provided do not include 4 cm as the correct radius for 64π surface area. Rechecking the calculation: 4πr² = 64π implies r² = 16, so r = 4 cm, which matches option C. Therefore, the original correct_answer 'C' is correct. The explanation was missing and is now provided. Difficulty is set to easy as this is a straightforward calculation.

The surface area of a rectangular box (cuboid) is calculated using the formula 2(lw + lh + wh), where l, w, and h are the length, width, and height respectively. For the box with dimensions 25 cm × 15 cm × 8 cm, the surface area is 2(25×15 + 25×8 + 15×8) = 2(375 + 200 + 120) = 2(695) = 1390 cm². Therefore, the area of cardboard needed to make the box is 1390 cm², which corresponds to option A.

The wall volume is calculated as 10 m × 4 m × 0.5 m = 20 m³. Since 10% of the volume is mortar, the volume occupied by bricks is 90% of 20 m³, which is 18 m³. Each brick measures 25 cm × 15 cm × 8 cm, which converts to 0.25 m × 0.15 m × 0.08 m = 0.003 m³ per brick. The number of bricks required is the brick volume divided by the volume of one brick: 18 m³ ÷ 0.003 m³ = 6,000 bricks. This matches option C. The initial confusion in the question's explanation about wall thickness being 5 m instead of 0.5 m was incorrect. The correct thickness is 0.5 m as stated in the question, leading to the correct answer of 6,000 bricks (option C).