Question

a student has a tennis ball can with a flat top and bottom containing three tightly fitting tennis balls. to the students surprise, the circumference of the top of the can is longer than the height of the can. the student wants to know if this fact can be explained without performing any measurements. how do you respond? let d denote the diameter of the tennis ball. choose the correct answer below a. the height is 3d. the circumference of the top of the can is 4d, which is greater than 3d b. the height is 3d. the circumference of the top of the can is 2πd which is approximately 6.28d, which is greater than 3d. c. the height is 3d. the circumference of the top of the can is πd which is approximately 3.14d, which is greater than 3d d. the height is 6d. the circumference of the top of the can is 2πd which is approximately 6.28d, which is greater than 6d

Explanation:

Step1: Determine the height of the can

Since there are three tightly - fitting tennis balls in the can and the diameter of each tennis ball is \(d\), the height of the can \(h = 3d\).

Step2: Calculate the circumference of the top of the can

The top of the can is a circle. The formula for the circumference of a circle is \(C=\pi\times d\) (where \(d\) is the diameter of the circle, and here the diameter of the top of the can is equal to the diameter of the tennis ball). Using \(\pi\approx3.14\), we have \(C = \pi d\approx3.14d\). But if we use the formula \(C = 2\pi r\) (where \(r=\frac{d}{2}\)), \(C=2\pi\times\frac{d}{2}=\pi d\). Another way, if we consider the formula \(C = 2\pi r\) (for a circle, and the diameter of the can is \(d\), so radius \(r=\frac{d}{2}\)), \(C = 2\pi r\), substituting \(r = \frac{d}{2}\) gives \(C=\pi d\approx3.14d\). But if we use the more accurate approximation \(\pi\approx3.14\), and we know that \(3.14d>3d\).

Answer:

B. The height is \(3d\). The circumference of the top of the can is \(2\pi d\) which is approximately \(6.28d\), which is greater than \(3d\).