Final answer:
To find the angle formed by a diagonal of a rectangular box with its base, calculate the base diagonal using the Pythagorean Theorem and then use the inverse tangent function with the height and base diagonal as arguments.
Step-by-step explanation:
The angle formed by a diagonal of the rectangular box with the base of the box can be found using the Pythagorean Theorem. In this case, the box has a length of 6, a width of 4, and a height of 5. The diagonal of the base can be calculated as the hypotenuse of a right triangle formed by the length and the width. According to the Pythagorean theorem, this diagonal (D) is √(6² + 4²) which is √(36 + 16) or √52. To find the angle between the box's base diagonal and the body diagonal (which includes the height), we need to use trigonometry, specifically the inverse tangent function (tan-1). So we have the length of the box's base diagonal (D) and the height (H), forming a right triangle with the body diagonal as the hypotenuse. The angle θ can be calculated using tan-1(H/D), where H is the height of the box and D is the length of the diagonal on the base. This equates to tan-1(5/√52). Evaluating this expression with a calculator gives us the angle θ formed by the body diagonal with the base of the box.