Can You Find The Volume Of Pyramids And Cones?

Explanation
The volume of a pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base of the pyramid is an equilateral triangle with a perimeter of 120 cm, which means each side is 40 cm. The height of the pyramid can be found using the Pythagorean theorem, where the slant height is the hypotenuse and half the base length is one of the legs. Solving for the height, we get h = sqrt(25^2 - 20^2) = 15 cm. The base area of an equilateral triangle can be calculated using the formula (sqrt(3)/4) * side^2, which gives us a base area of (sqrt(3)/4) * 40^2 = 400sqrt(3) cm^2. Plugging in these values into the volume formula, we get V = (1/3) * 400sqrt(3) * 15 = 8000 cm^3.

Explanation
The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this case, the base has a diameter of 30 mm, so the radius is half of that, which is 15 mm. The height is given as 25 mm. Plugging these values into the formula, we get V = (1/3)π(15^2)(25) = 5888 mm^3.

Explanation
The volume of a cone can be calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. In this case, the radius is half of the diameter, so it is 3.5 cm. Plugging in the values, we get V = (1/3)π(3.5)^2(9) = 115.4 cm^3. Therefore, the cup can hold 115.4 cm^3 of water.

Explanation
The paperweight is in the shape of a square pyramid, which means it has a square base. The side length of the square base is given as 7 cm. The volume of a pyramid is calculated by multiplying the area of the base by the height and then dividing by 3. The area of a square is calculated by multiplying the side length by itself. In this case, the area of the base is 7 cm * 7 cm = 49 cm^2. The height of the paperweight is given as 7.5 cm. Plugging these values into the formula, we get (49 cm^2 * 7.5 cm) / 3 = 122.5 cm^3. Therefore, the paperweight can hold 122.5 cm^3 of water.

Explanation
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. Since the diameter is 4 m, the radius is 2 m. We can rearrange the formula to solve for h: h = (3V)/(πr^2). Plugging in the given volume of 23 m^2 and radius of 2 m, we get h = (3 * 23) / (π * 2^2) = 69 / (4π) ≈ 5.9 m. Therefore, the slant height of the cone is 5.9 m.

Explanation
The volume of a square pyramid can be calculated using the formula V = (1/3) * base area * height. In this case, the base area is equal to the side length squared (24 cm * 24 cm) and the height is equal to the slant height (15 cm). Plugging these values into the formula, we get V = (1/3) * 24 cm * 24 cm * 15 cm = 1728 cm^3. Therefore, the correct answer is 1728 cm^3.

Explanation
The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. In this question, the diameter of the base is given as 12 cm, so the radius would be half of that, which is 6 cm. The slant height of the cone is given as 10 cm, which is the height of the cone. Plugging these values into the formula, we get V = (1/3)π(6^2)(10) = 301.4 cm^3.

Explanation
The volume of a regular hexagonal pyramid can be calculated using the formula V = (1/3) * A * h, where A is the area of the base and h is the height of the pyramid. In this case, the base is a regular hexagon, so its area can be calculated using the formula A = (3√3/2) * s^2, where s is the length of one side of the hexagon. The length of one side of the hexagon can be calculated using the distance from the center of the base to the midpoint of each side, which is given as 1.04 cm. Therefore, s = 2 * 1.04 cm = 2.08 cm. Plugging these values into the formulas, we get A ≈ 10.392 cm^2 and V ≈ 1.87 cm^3.

Explanation
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the volume is given as 16.75 cm^3 and the height and diameter are the same. Since the height and diameter are the same, the height can be used as the diameter. We can rearrange the formula to solve for the radius: r = √((3V)/(πh)). Plugging in the given values, we get r = √((3*16.75)/(π*2)). Simplifying this expression gives r ≈ 2 cm. Therefore, the radius of the cone is 2 cm.