To determine the longest fishing pole that Bruce can ship in a cylindrical container with a diameter of 13 inches and a height of 84 inches, we need to consider the constraints of the cylinder.
The diameter of the cylinder is 13 inches, which means the radius is:
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{13 \text{ inches}}{2} = 6.5 \text{ inches} \]
Now, to find the longest fishing pole that can fit inside the cylinder, we can visualize the situation. The longest pole that can fit inside a cylinder will touch the sides at its endpoints and will also fit within the height of the cylinder. The longest line that can be drawn inside a cylinder is along its diagonal from one corner of the base to the top of the opposite edge.
To calculate this, we can use the Pythagorean theorem in three dimensions. In this case, we form a right triangle where:
- One leg is the height of the cylinder (84 inches)
- The other leg is the diameter of the base of the cylinder (which would be the diagonal of the circle formed by the base).
- Calculate the diagonal of the cylinder's base: The diagonal across the base of the cylinder can be calculated as:
\[ d = \sqrt{(2 \times \text{radius})^2 + (2 \times \text{radius})^2} = \sqrt{(13)^2 + (13)^2} = \sqrt{169 + 169} = \sqrt{338} = 13 \sqrt{2} \text{ inches} \]
- Now we can calculate the length of the pole fitting in the cylinder: Using the height and the diameter of the base:
\[ L = \sqrt{(\text{Height})^2 + (\text{Diameter})^2} = \sqrt{(84)^2 + (13)^2} = \sqrt{7056 + 169} = \sqrt{7225} = 85 \text{ inches} \]
Therefore, the longest fishing pole that Bruce can ship to his brother is 85 inches.
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