Final answer:
To find the length of the rubber band that fits snugly around three cylindrical canisters with a radius of 1.5 inches, we calculate the perimeter of the equilateral triangle formed by the canisters and add the lengths of the three arcs on the outside. The total length of the rubber band is approximately 23.13 inches.
Step-by-step explanation:
Jane has three cylindrical canisters each with a radius of 1.5 inches, and she needs to find the length of the rubber band to hold them together snugly. To do this, we envision the canisters touching each other in a line, forming a triangle within the circle created by their outer sides. The canisters will act as vertices of an equilateral triangle, where the sides of the triangle are equal to the diameters of the canisters. Since the radius is 1.5 inches, the diameter is 3 inches.
The rubber band will wrap around this triangle, so its length will be equal to the perimeter of the triangle formed. For an equilateral triangle, the perimeter is just three times the length of one side. The perimeter P is calculated as P = 3 × (2 × 1.5 inches) = 9 inches. However, this is only the length around the triangle formed by the centers of the cylinders, not around the outside. To find the complete length, we need to add the circumferences of the rounded parts where the canisters touch. Each of these arc lengths is one-third of the total circumference of a circle with radius 1.5 inches. The circumference C of a full circle is C = 2πr so the arc length for one-third of the circumference is 2πr/3. There are three such arcs, so we need to multiply this by 3. The total length L of the rubber band is then L = P + 3 × (2πr/3).
Therefore, the total length of the rubber band will be 9 inches + 3 × (π × 1.5 inches). Simplifying this gives us L = 9 inches + 14.13 inches, which is approximately 23.13 inches.