Finding Cone Diameter from Given CSA and Slant Height

This problem requires us to calculate the diameter of the base of a cone when we know its Curved Surface Area (CSA) and slant height. We will use the standard formula for the CSA of a cone and solve for the radius first, then find the diameter.

Understanding the Cone Formulas

Key formulas related to a cone:

• Curved Surface Area (CSA) of a cone: $\text{CSA} = \pi \times r \times l$
• Radius ($r$), Slant Height ($l$), and Height ($h$) relationship: $l^2 = r^2 + h^2$ (Though height is not needed here)
• Diameter ($d$) of the base: $d = 2 \times r$

Given Information

We are provided with:

• Curved Surface Area (CSA) = 165 cm²
• Slant Height ($l$) = 10 cm

Step-by-Step Calculation

1. Calculate the Radius (r):

   We use the CSA formula and substitute the given values:

   $$ \text{CSA} = \pi \times r \times l $$

   $$ 165 \, \text{cm}^2 = \pi \times r \times 10 \, \text{cm} $$

   To find the radius ($r$), we rearrange the formula:

   $$ r = \frac{165 \, \text{cm}^2}{\pi \times 10 \, \text{cm}} $$

   Let's use the value of $\pi \approx \frac{22}{7}$:

   $$ r = \frac{165}{\frac{22}{7} \times 10} $$

   $$ r = \frac{165 \times 7}{22 \times 10} $$

   $$ r = \frac{1155}{220} \, \text{cm} $$

   Simplifying the fraction:

   $$ r = \frac{1155 \div 5}{220 \div 5} = \frac{231}{44} \, \text{cm} $$

   $$ r = \frac{231 \div 11}{44 \div 11} = \frac{21}{4} \, \text{cm} $$

   Converting the fraction to a decimal:

   $$ r = 5.25 \, \text{cm} $$

2. Calculate the Diameter (d):

   The diameter is twice the radius.

   $$ d = 2 \times r $$

   $$ d = 2 \times 5.25 \, \text{cm} $$

   $$ d = 10.5 \, \text{cm} $$

Conclusion

The diameter of the base of the cone is 10.5 cm.