Final answer:
Using the Pythagorean theorem in three dimensions, we find that the minimum whole-number length of the suitcase to fit a 34-inch baseball bat, given its height and depth of 17 inches and 11 inches respectively, is 28 inches.
Step-by-step explanation:
The question is asking to find the minimum length of a suitcase that can fit a 34-inch baseball bat, considering the suitcase's other dimensions are 17 inches in height and 11 inches in depth. To determine the minimum length, we will use the Pythagorean theorem as this is a three-dimensional problem involving finding the diagonal of a rectangular prism (the suitcase).
First, let us consider the two dimensions given: 17 inches (height) and 11 inches (depth). The baseball bat needs to fit diagonally because its length exceeds each of these dimensions. We'll use the Pythagorean theorem in three
dimensions to find the length of the diagonal:
c² = a² + b² + l²
where c is the diagonal we're trying to find (at least 34 inches for the baseball bat), a is the height (17 inches), b is the depth (11 inches), and l will be the length of the suitcase we want to find.
Let's calculate:
34² = 17² + 11² + l²
1156 = 289 + 121 + l²
1156 = 410 + l²
l² = 1156 - 410
l² = 746
Now, we need to find the square root of 746 to find l. The square root of 746 is approximately 27.31. Since we need a whole number that is at least the length of the diagonal, and the length of the suitcase must be an integer, the minimum whole number length of the suitcase must be 28 inches.
Therefore, the minimum whole-number length the suitcase must be to fit the bat is 28 inches.