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The cost of polishing the total surface area of a solid cone at ₹0.50 per cm² is ₹5632, and the circumference of its base is 176 cm. What is the height of the cone? Use [tex]π = \frac{22}{7}[/tex].

Answer :

To find the height of the cone, we first need to understand the surface area of a cone and how it relates to the given details.

Step-by-Step Solution:


  1. Understand the Problem:


    • The cost of polishing the surface area at ₹0.50 per cm² is ₹5632.

    • Circumference of the base of the cone is given as 176 cm.



  2. Determine the Total Surface Area (TSA):


    • Since the cost is ₹5632, the total surface area can be found by dividing the total cost by the cost per cm²:

      [tex]\text{Total Surface Area} = \frac{5632}{0.50} = 11264 \text{ cm}^2[/tex]



  3. Surface Area of a Cone:


    • The total surface area (TSA) of a cone is the sum of its curved surface area (CSA) and the area of the base.

    • Mathematically, [tex]\text{TSA} = \pi r l + \pi r^2[/tex], where [tex]r[/tex] is the radius and [tex]l[/tex] is the slant height.



  4. Determine the Radius:


    • The base is a circle, and its circumference [tex]C = 2\pi r = 176 \text{ cm}[/tex].

    • Solving for [tex]r[/tex]:

      [tex]2 \pi r = 176[/tex]

      [tex]2 \times \frac{22}{7} \times r = 176[/tex]

      [tex]r = \frac{176 \times 7}{2 \times 22} = 28 \text{ cm}[/tex]



  5. Calculate the Slant Height ([tex]l[/tex]):


    • Use TSA formula to solve for [tex]l[/tex]:

      [tex]\pi r l + \pi r^2 = 11264[/tex]

      [tex]\frac{22}{7} \times 28 \times l + \frac{22}{7} \times 28^2 = 11264[/tex]

      [tex]\frac{22}{7} \times 28 (l + 28) = 11264[/tex]

      [tex]88 (l + 28) = 11264[/tex]

      [tex]l + 28 = \frac{11264}{88} = 128[/tex]

      [tex]l = 128 - 28 = 100 \text{ cm}[/tex]



  6. Find the Height ([tex]h[/tex]):


    • The slant height [tex]l[/tex], radius [tex]r[/tex], and height [tex]h[/tex] are related by the Pythagorean theorem: [tex]l^2 = r^2 + h^2[/tex].

    • Substitute values:

      [tex]100^2 = 28^2 + h^2[/tex]

      [tex]10000 = 784 + h^2[/tex]

      [tex]h^2 = 10000 - 784 = 9216[/tex]

      [tex]h = \sqrt{9216} = 96 \text{ cm}[/tex]




Hence, the height of the cone is [tex]96 \text{ cm}[/tex].

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Rewritten by : Barada