2070.61 – Big Round Dinosaur Buttons

The remnants we have of dinosaurs are few and make it difficult to know what they actually looked like beyond their skeletal forms and feathery exteriors. It is a little-known fact that many wore coats in the winter, and those coats had large wooden buttons. Intelligent troodons ran a button factory in which they developed a system for cutting four holes, each with a radius of y inches, out of a circular piece of wood with a radius of x inches. (There was a steady supply of such circular pieces; it is not understood where they came from.) Assuming the troodons’ motor control was sound and the holes did not overlap, write a formula for the area of remaining wood, and evaluate the formula if x = 10 inches and y = 2 inches. (Use 3.14 for $\pi$ )

Solution

The area of the original plywood is $\pi x^2$ . Each hole has an area $\pi y^2$ ; there are four of them. Thus the remaining area is $\pi x^2 – 4\pi y^2$ .

If $x = 10$ and $y = 2$ and we use $\pi \approx 3.14$ , we get $3.14 (100) - 4(3.14)(4) = 263.76.$

Interesting extension: If you're going to cut four round buttons out of that circular piece of wood with radius 10 inches, what's the maximum radius for the buttons?