Mathematics

The highest common factor (HCF) of 36 and 84 is 12, as 12 is the largest number that divides both 36 and 84 without leaving a remainder.

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.

To simplify 1095/1168, we find the greatest common divisor (GCD) of 1095 and 1168, which is 43. Dividing numerator and denominator by 43 gives 25/26, making option D correct.

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.

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 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 $4\pi r^2=4\pi (5{)}^2=100\pi$.

To find the G.C.D of decimal numbers, convert them to integers by multiplying by a power of 10 that removes the decimals. Here, multiply each by 100: 1.08 = 108, 0.36 = 36, 0.9 = 90. The G.C.D of 108, 36, and 90 is 18. Dividing back by 100 gives 0.18. Therefore, the G.C.D of 1.08, 0.36, and 0.9 is 0.18, which corresponds to option C.

The greatest common divisor (GCD) of 80, 144, and 368 is 16. This means each can should contain 16 liters of a single drink to maximize the volume per can without mixing drinks. The number of cans required is calculated by dividing each drink quantity by 16 and summing the results: (80/16) + (144/16) + (368/16) = 5 + 9 + 23 = 37 cans. Therefore, the least number of cans required is 37, which corresponds to option B.

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 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.