Cone Curved Surface Area Ratio Calculation
We need to find the ratio of the curved surface areas of two cones where their base diameters are equal, and their slant heights are in the ratio 3:4.
Key Information
- Two cones
- Equal base diameters (implies equal radii)
- Ratio of slant heights ($l_1 : l_2$) = 3 : 4
Curved Surface Area Formula
The formula for the curved surface area (CSA) of a cone is:
$CSA = \pi r l$
where $r$ is the radius of the base and $l$ is the slant height.
Step-by-Step Solution
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Let the radius of the base for both cones be $r$, since their diameters are equal.
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Let the slant heights of the two cones be $l_1$ and $l_2$.
We are given the ratio: $\frac{l_1}{l_2} = \frac{3}{4}$
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Calculate the curved surface area for the first cone ($CSA_1$):
$CSA_1 = \pi r l_1$
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Calculate the curved surface area for the second cone ($CSA_2$):
$CSA_2 = \pi r l_2$
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Find the ratio of their curved surface areas:
$\frac{CSA_1}{CSA_2} = \frac{\pi r l_1}{\pi r l_2}$
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Cancel out the common terms ($\pi$ and $r$):
$\frac{CSA_1}{CSA_2} = \frac{l_1}{l_2}$
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Substitute the given ratio of slant heights:
$\frac{CSA_1}{CSA_2} = \frac{3}{4}$
Therefore, the ratio of their curved surface areas is 3:4.
Similar Questions
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Important Questions from 3-D Mensuration
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
The length and breadth of a cuboid are 50 cm and 75 cm, respectively. If the length of the diagonal is $17\sqrt{29}$ cm, then what is the volume (in $cm^{3}$) of the cuboid?
