Challenge A cylindrical vase has height 17 in . and radius 3 in . Find the exact surface area of the vase in terms of pi \( \pi \). Suppose a second vase has double the radius, but the same surface area. What is the height of this vase?

To find the surface area of the cylindrical vase, we use the formula: \[ \text{Surface Area} = 2\pi r(h + r) \] where \( r \) is the radius and \( h \) is the height. Plugging in the values, we have: \[ \text{Surface Area} = 2\pi (3)(17 + 3) = 2\pi (3)(20) = 120\pi \text{ in}^2. \] Now, for the second vase with double the radius, the radius \( r' = 6 \, \text{in} \) and the surface area remains \( 120\pi \). The formula again gives us: \[ 120\pi = 2\pi(6)(h' + 6) \] Dividing both sides by \( 2\pi \): \[ 60 = 6(h' + 6) \] Simplifying that: \[ 10 = h' + 6 \] Thus, subtracting 6 from both sides: \[ h' = 4 \, \text{in}. \] There you have it! The height of the second vase is 4 inches. Did you know that the concept of cylindrical shapes dates back to ancient civilizations? The Egyptians utilized these shapes in their architecture, like the pyramids, where cylindrical columns were quite common! Their understanding of geometry certainly laid the groundwork for modern mathematics. In practical terms, understanding the surface area of a vase is super useful! If you're planning to paint or decorate it, you'll want to know that area to buy enough materials. Likewise, if you're filling the vase with flowers or decorative items, it helps you gauge how much to buy for that perfect look!