Mensuration

The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base, h is the height of the cylinder, and π is the constant pi (approximately 3.14). Therefore, the volume of a cylinder with radius 9 cm and height 18 cm is π(9²)(18) = 4536π cm³.

Area of a circle = π * radius² = π * 7 cm² = 154 cm²

Area of a sector = (π/360) * radius² * central angle = (π/360) * 10 cm² * 60° = 30π cm²

Area of a segment = (1/2) * (radius² - height²) = (1/2) * (12 cm² - 8 cm²) = 96π cm²

The volume of a sphere is given by (\frac{4}{3}\pi r^3). Substituting (r = 5) cm, we get (\frac{4}{3}\pi (5)^3 = 523.6) cm³.

The volume of a cylinder is given by (\pi r^2 h). Substituting (r = 3) cm and (h = 8) cm, we get (\pi (3)^2 (8) = 75.4) cm³.

The volume of a cone is given by (\frac{1}{3}\pi r^2 h). Substituting (r = 4) cm and (h = 10) cm, we get (\frac{1}{3}\pi (4)^2 (10) = 33.5) cm³.

The surface area of a cube is given by (6s^2), where (s) is the length of one side. Solving for (s), we get (s = \sqrt{\frac{A}{6}}). Substituting (A = 150) cm², we get (s = \sqrt{\frac{150}{6}} = 5) cm.

The surface area of a sphere is given by (4\pi r^2). Solving for (r), we get (r = \sqrt{\frac{A}{4\pi}}). Substituting (A = 144) cm², we get (r = \sqrt{\frac{144}{4\pi}} = 4) cm.

The surface area of a cylinder is given by (2\pi r h + 2\pi r^2). Solving for (h), we get (h = \frac{A - 2\pi r^2}{2\pi r}). Substituting (A = 300) cm², (r = 5) cm, and (\pi \approx 3.14), we get (h = \frac{300 - 2\pi (5)^2}{2\pi (5)} = 10) cm.

The area of a circle is calculated using the formula: $$Area = \pi r^2$$ Substituting the given radius, we get: $$Area = \pi \times 7^2$$ $$Area = \pi \times 49$$ $$Area = 154 cm^2$$

The volume of a cylinder is calculated using the formula: $$Volume = \pi r^2 h$$ Substituting the given radius and height, we get: $$Volume = \pi \times 5^2 \times 10$$ $$Volume = \pi \times 25 \times 10$$ $$Volume = 250 cm^3$$

Cylinder volume = pi * r^2 * h = 144. Since r = h = radius of sphere (R), pi * R^3 = 144. Hemisphere volume = 2/3 * pi * R^3. Substituting pi * R^3 = 144, volume = 2/3 * 144 = 96.

For a square of side 10 cm, the incircle radius is 5 cm and the circumcircle radius is 5*sqrt(2) cm. The ratio of areas is (pi*5^2) / (pi*(5*sqrt(2))^2) = 25 / 50 = 1/2.

r+h=15. TSA = 2*pi*r(r+h) = 660. 2*pi*r*15 = 660. r = 660 / (30*pi) = 22/pi. h = 15 - 22/pi. Volume = pi*r^2*h = pi*(22/pi)^2 * (15 - 22/pi) = (484/pi) * (15 - 22/pi) = 7260/pi - 484*22/pi^2. Using pi approx 22/7, V = 7260/(22/7) - 10648/(484/49) = 2310 - 1078 = 1232.