Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm . If they make the diameter 13 cm , what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth.

To find the slant height of the cone, we can use the Pythagorean Theorem. First, we need to determine the radius of the cone, which is half the diameter. So the radius is 13 cm / 2 = 6.5 cm. The formula is \( a^2 + b^2 = c^2 \), where \( a \) is the radius (6.5 cm), \( b \) is the height (6 cm), and \( c \) is the slant height. Plugging in the values gives us \( 6.5^2 + 6^2 = c^2 \) or \( 42.25 + 36 = c^2 \). Now, \( 78.25 = c^2 \) leads to \( c = \sqrt{78.25} \). This calculates to approximately 8.8 cm when rounded to the nearest tenth. In real-world applications, understanding the geometry of cones is crucial for various industries, including food packaging and chemical filtering. For example, when designing efficient filters, knowing the slant height helps ensure that the cones not only fit the desired liquid containment systems but also optimize flow rates and material usage. Such practical applications highlight why mastering basic geometric principles is essential for engineers and scientists alike!