Final answer:
To determine the dimensions of the largest inscribed rectangle, we must find where the curve y = x^4 - 20 intersects y = 10. Through optimization techniques in calculus, the width and height at the point of intersection can be computed, but given answer choices, we can eliminate option (c) immediately as it exceeds the bounds.
Step-by-step explanation:
To find the dimensions of the largest rectangle that can be inscribed in the border of the region bounded by the curves y = 10 and y = x4 - 20, we need to understand that the upper bound for the rectangle's height is given by y = 10 and the lower bound is the curve y = x4 - 20.
The region where the rectangle can be inscribed must satisfy the condition 0 ≤ x ≤ 20, as stated by scaling the x and y axes with the maximum x and y values.
Since the dimensions are not directly given, we need to find the maximum area that can be attained under these curves. The maximum area for an inscribed rectangle under a curve can be found using calculus, specifically by utilizing the method of optimization with constraints.
The answer options suggest specific dimensions for the rectangle. The rectangle's width is constrained by the x-values where the curve y = x4 - 20 intersects y = 10. This intersection occurs where x4 - 20 = 10, which means x4 = 30.
By finding the x-value where this condition is true, we can then calculate the width and height of the rectangle at that point, which would yield the dimensions of the largest possible inscribed rectangle. However, option (c) (20, 400) is clearly incorrect as the x-value exceeds the range and the corresponding y-value goes beyond the upper bound of the rectangle's height. The other options can be tested to check if they meet the conditions required.