Explanation:
The height (h) of a cone can be calculated using the Pythagorean theorem, as the slant height (l), radius (r), and height form a right triangle. The relationship is given by:
l² = r² + h²,
where:
– l is the slant height,
– r is the radius of the base,
– h is the height of the cone.

From the problem:
– The slant height (l) is 26 cm,
– The diameter of the base is 20 cm, so the radius (r) is:
r = Diameter / 2 = 20 / 2 = 10 cm.

Substitute the values of l = 26 cm and r = 10 cm into the formula:
l² = r² + h².

Rearrange to solve for h²:
h² = l² – r².

Substitute the values:
h² = 26² – 10²,
h² = 676 – 100,
h² = 576.

Take the square root of both sides:
h = √576,
h = 24 cm.

Thus, the height of the cone is 24 cm, which corresponds to option a) 24.

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Explanation:
The curved surface area (CSA) of a right circular cone is given by the formula:
CSA = πrl,
where:
– r is the radius of the base,
– l is the slant height of the cone.

From the problem:
– The radius (r) of the base is 7 cm,
– The slant height (l) is 10 cm.

Substitute the values of r = 7 cm and l = 10 cm into the formula:
CSA = πrl.

Using π ≈ 22/7 for calculation:
CSA = (22/7) × 7 × 10.

Simplify step by step:
CSA = 22 × 10,
CSA = 220 cm².

Thus, the curved surface area of the cone is 220 cm², which corresponds to option b) 220.

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Explanation:
The roller is a cylinder, and the area it covers in one complete revolution is equal to its curved surface area (CSA). The formula for the CSA of a cylinder is:
CSA = 2πrh,
where:
– r is the radius of the circular base,
– h is the length (or height) of the cylinder.

From the problem:
– The diameter of the roller is 84 cm, so the radius (r) is:
r = Diameter / 2 = 84 / 2 = 42 cm = 0.42 m (converted to meters),
– The length (h) of the roller is 120 cm = 1.2 m (converted to meters).

Substitute the values of r = 0.42 m and h = 1.2 m into the formula:
CSA = 2πrh.

Using π ≈ 22/7 for calculation:
CSA = 2 × (22/7) × 0.42 × 1.2.

Simplify step by step:
CSA = 2 × (22/7) × 0.504,
CSA = 2 × 22 × 0.072,
CSA = 3.168 m².

This is the area covered by the roller in one complete revolution. Since the roller takes 500 revolutions to level the playground, the total area of the playground is:
Total Area = CSA × Number of Revolutions,
Total Area = 3.168 × 500,
Total Area = 1584 m².

Thus, the area of the playground is 1584 m², which corresponds to option a) 1584.

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