Question

The radius of cone is 10 cm. The ratio of curved surface area and the total surface area of cone is 4: 5. Find the slant height of the cone.

• A. 20 cm
• B. 40 cm
• C. 30 cm
• D. 50 cm

Answer 1

Answer:

B. 40 cm

Key Concepts for Solids: Cones

• A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

• Radius (r): The distance from the center of the base to any point on the circumference of the base.

• Height (h): The perpendicular distance from the apex to the center of the base.

• Slant Height (l): The distance from the apex to any point on the circumference of the base. It is related to the radius and height by the Pythagorean theorem: l² = r² + h².

• Curved Surface Area (CSA): The area of the slanted surface of the cone. Formula: CSA = πrl.

• Total Surface Area (TSA): The sum of the curved surface area and the area of the base. Formula: TSA = CSA + Base Area = πrl + πr² = πr(l + r).

Problem Breakdown and Solution Approach

• Given:

  • Radius of the cone (r) = 10 cm

  • Ratio of Curved Surface Area (CSA) to Total Surface Area (TSA) = 4:5

• To Find: Slant height (l) of the cone.

• Using the Formulas:

  • CSA = πrl

  • TSA = πr(l + r)

• Applying the Ratio:

  CSA / TSA = (πrl) / (πr(l + r)) = 4/5

• Simplifying the Equation:

  l / (l + r) = 4/5

• Cross-multiplication:

  5l = 4(l + r)

  5l = 4l + 4r

• Solving for l:

  5l - 4l = 4r

  l = 4r

• Substituting the value of r:

  l = 4 × 10 cm

  l = 40 cm