6) Find the radius of a cone whose volume is \( 33.49 \mathrm{~cm}^{3} \) and height 2 cm.

To find the radius of a cone given its volume and height, you can use the formula for the volume of a cone: \[ V = \frac{1}{3} \pi r^2 h \] Where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. Rearranging the formula to solve for \( r \): \[ r^2 = \frac{3V}{\pi h} \] Plugging in the values \( V = 33.49 \, \text{cm}^3 \) and \( h = 2 \, \text{cm} \): \[ r^2 = \frac{3 \times 33.49}{\pi \times 2} \approx 15.92 \] Taking the square root gives: \[ r \approx \sqrt{15.92} \approx 3.98 \, \text{cm} \] So, the radius of the cone is approximately \( 3.98 \, \text{cm} \). In ancient civilizations, conical shapes were not only used in architecture, such as pyramids, but also in vessels and religious artifacts. The geometry of cones has fascinated mathematicians for centuries, leading to developments in calculus and even applications in modern physics, such as in understanding structures and fluid dynamics. For practical applications, understanding the properties of cones can be pivotal in fields like engineering and design. For example, when creating a funnel or a conical hat, knowing the radius can determine the volume of materials needed. Plus, a well-measured cone can help reduce waste and enhance functionality, making it a fun element in DIY projects too!