The main diameter that a tennis ball, as characterized by the international Tennis Federation, is at the very least 2.575 inches and also at many 2.700 inches. Tennis balls are sold in cylindrical containers that contain three balls each. To model the container and also the balls in it, we will certainly assume that the balls are 2.7 customs in diameter and that the container is a cylinder the inner of which actions 2.7 inches in diameter and $3 imes 2.7 = 8.1$ inch high.

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Lying ~ above its side, the container passes through an X-ray scanner in an airport. If the product of the container is opaque to X-rays, what overview will appear? through what dimensinos?

If the material of the container is partly opaque to X-rays and the material of the balls is completely opaque come X-rays, what will certainly the summary look like (still suspect the have the right to is lie on its side)?

The *central axis* of the container is a line the passes v the centers of the top and bottom. If one cut the container and balls by a aircraft passing through the central axis, what walk the intersection of the aircraft with the container and also balls look at like? (The intersection is likewise called a *cross section*. Imagine placing the cut surface top top an squid pad and then stamping a piece of paper. The stamped image is a snapshot of the intersection.)

If the can is reduced by a plane parallel come the central axis, but at a street of 1 inch from the axis, what will certainly the intersection that this aircraft with the container and also balls look at like?

If the deserve to is cut by a airplane parallel to one finish of the can—a horizontal plane—what are the possible appearances that the intersections?

A cross-section by a horizontal plane at a height of $1.35 + w$ inches native the bottom is made, with $0 lt w lt 1.35$ (so the bottom ball is cut). What is the area the the section of the cross section inside the container but outside the tennis ball?

Suppose the deserve to is reduced by a aircraft parallel to the main axis however at a street of $w$ inches from the axis ($0lt wlt 1.35$). What fractional part of the cross ar of the container is within of a tennis ball?

## IM Commentary

This job is motivated by the source of the volume formula for the sphere. If a round of radius 1 is enclosed in a cylinder that radius 1 and also height 2, then the volume not occupied by the round is same to the volume that a “double-naped cone” v vertex at the facility of the sphere and bases same to the bases of the cylinder. This deserve to be viewed by slicing the number parallel to the base of the cylinder and also noting the areas of the annular slices consisting of parts of the volume that room *inside* the cylinder yet *outside* the sphere space the same as the locations of the slices that the double-naped cone (and applying Cavalieri’s Principle). This virtually magical fact about slices is a manifestation that Pythagorean Theorem. We watch it at occupational in component 6 of this task. The various other parts the the task are exercises in 3D-visualization, which develop up the spatial sense essential to occupational on part 6 v understanding. The visualization required here is offered in calculus, in link with procedures for calculating volumes by assorted slicing procedures.

Submitted by James Madden on behalf of the entrants in the Louisiana Math and also Science Teacher academy On-Ramp.

## Solution

The shadow is a rectangle measure up 2.7 inches by 8.1 inches.

The shadow is a irradiate rectangle (2.7 × 8.1 inches) with 3 disks inside. The looks prefer a web traffic light:

The picture is comparable to the previous one, but now just the outlines room seen:

The intersection with the container is a narrower rectangle. The intersections through the balls are smaller sized circles. Due to the fact that each ball touches the container along its entirety “equator,” the circles must touch the lengthy sides of the rectangle:

The intersections room two concentric circles, other than when $w = 0, 2.7, 5.4, 8.1$ and also when $w = 1.35, 4.05, 6.75$. In the former case, we view a circle (from the container) and also a allude (where the plane touches a sphere). In the last case, we view a single circle corresponding to a ar where the equator the a sphere touches the container.

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The intersection the the aircraft with the internal of the *container* is a disc of radius 1.35 inches. The area is $pi (1.35)^2 ext in^2$. The intersection v the *ball* is a smaller sized disk that is consisted of in the an initial disk. The radius $r$ the the smaller sized disk is the square root of $(1.35)^2 - w^2$, together we watch from the diagram listed below depicting the intersection that a aircraft through the central axis of the container through the bottom ball. Thus, the area of the smaller disk is $pi ( (1.35)^2 - w^2 )$. Accordingly, the area within the bigger disk yet outside the smaller is $pi w^2$, detailed that $0 leq w leq 1.35$. (It is notable that the radius the the ball does not appear explicitly in the expression because that this annular area.)

Referring to trouble d), we check out that we wish to find the proportion of the total area of three congruent disks to the area that a rectangle, among whose dimensions is equal to the diameter the the disks. The same snapshot used in the vault problem, but interpreted as a watch from one end of the container, offers us the radius the the little disks — namely,$sqrt(1.35)^2 - w^2$, therefore the total area the the disks is $3 pi ( (1.35)^2 - w^2 )$. The area the the rectangle is $(8.1) 2 sqrt(1.35)^2 - w^2$. So, the proportion is

$$frac3 pi ( (1.35)^2 - w^2 )(8.1) 2 sqrt(1.35)^2 – w^2 = fracpi sqrt (1.35)^2 - w^25.4$$The official diameter the a tennis ball, as identified by the international Tennis Federation, is at the very least 2.575 inches and at most 2.700 inches. Tennis balls are sold in cylindrical containers that contain three balls each. To version the container and the balls in it, we will assume the the balls room 2.7 inch in diameter and that the container is a cylinder the internal of which steps 2.7 inches in diameter and also $3 imes 2.7 = 8.1$ inch high.

Lying on its side, the container passes with an X-ray scanner in an airport. If the product of the container is opaque to X-rays, what summary will appear? v what dimensinos?

If the material of the container is partly opaque to X-rays and the product of the balls is totally opaque to X-rays, what will certainly the overview look choose (still assuming the deserve to is lying on its side)?

The *central axis* the the container is a line that passes v the centers that the top and bottom. If one cut the container and balls by a plane passing v the main axis, what does the intersection of the aircraft with the container and balls look at like? (The intersection is additionally called a *cross section*. Imagine placing the reduced surface ~ above an ink pad and also then stamping a piece of paper. The stamped picture is a photo of the intersection.)

If the deserve to is cut by a airplane parallel come the main axis, however at a distance of 1 inch from the axis, what will the intersection that this aircraft with the container and also balls look at like?

If the can is reduced by a airplane parallel to one finish of the can—a horizontal plane—what space the feasible appearances of the intersections?

A cross-section by a horizontal aircraft at a elevation of $1.35 + w$ inches native the bottom is made, through $0 lt w lt 1.35$ (so the bottom round is cut). What is the area that the portion of the cross ar inside the container yet outside the tennis ball?

Suppose the deserve to is reduced by a airplane parallel to the central axis however at a street of $w$ inches indigenous the axis ($0lt wlt 1.35$). What fractional part of the cross section of the container is within of a tennis ball?