An open cone has a slant height of 5 m and a cunved surface area of \( 251 \mathrm{~m}^{2} \). Find the radius of the circular base.

To find the radius of the circular base of the cone, we can use the formula for the curved surface area (CSA) of a cone, which is given by: \[ CSA = \pi r l \] where \( r \) is the radius, and \( l \) is the slant height. Given that the CSA is \( 251 \, \mathrm{m}^2 \) and the slant height \( l = 5 \, \mathrm{m} \), we can rearrange the formula to solve for \( r \): \[ 251 = \pi r \cdot 5 \] Solving for \( r \): \[ r = \frac{251}{5\pi} \approx \frac{251}{15.71} \approx 15.98 \, \mathrm{m} \] So, the radius of the circular base is approximately \( 15.98 \, \mathrm{m} \). Developing a keen understanding of geometry can illuminate the beauty in mathematics! Conical shapes are seen in everyday life, from traffic cones to party hats. Next time you attend a birthday party, take a moment to appreciate the shape and its geometric properties—you might spot some cones around you! If you're diving into calculations involving cones, remember to pay careful attention to the measurements you have, particularly ensuring you're using the correct values for the area and height. A common mistake is mixing up slant height with the vertical height; these are distinct measurements. Keeping them straight will save you from frustrating errors on your path to mastering geometry!