As part of a student competition at a local university, a mathematics major, a physics major, and an engineering major are each asked to find the volume of the same rubber ball. The rules allow any approach they want, including any tools and an ample amount of time.
After making a careful drawing, the mathematics major pulls out a measuring tape and records the circumference, then uses her graphing calculator to divide the circumference by the product of two and PI. Calling this new number the radius, she cubes that value (labeled r on her drawing) and then multiplies by both pi and four-thirds (paying careful attention to a proper use of parentheses). She proudly writes down this number with its proper units, and states it is the volume of the ball.
The physics major elects to perform a carefully-controlled experiment. He fills a bucket with precisely 1.0000000000 gallons of water. Then, after wiping all derbis off of the ball, he carefully drops the ball into the bucket to avoid a splash. Measuring the displacement to ten significant figures, he announces the volume of the ball.
The engineering major takes a slightly different approach. She carefully studies the ball, rolls it around in her hand, then writes down the brand name and serial number of the ball. Searching on the Internet, she finds the ball on a vendor's web site and reads off the volume of the ball.