Final answer:
To find k, the value that allows a 58 unit diagonal of a rectangle to have an aspect ratio of 15:8, use the Pythagorean theorem and set up an equation involving x. Solve for x and use it to find the length and width of the rectangle. Calculate the diagonal length and round k to the nearest tenth.
Step-by-step explanation:
To find the value of k that allows the 58 unit diagonal of a rectangle to have an aspect ratio of 15:8, we can use the Pythagorean theorem. The formula for the diagonal of a rectangle is d = sqrt(l^2 + w^2), where d is the diagonal length, l is the length, and w is the width. Since the aspect ratio is 15:8, we can set up the equation (15x)^2 + (8x)^2 = 58^2 to represent the diagonal length in terms of x, the common factor for the length and width. Solving this equation, we find x ≈ 1.2. Therefore, the width of the rectangle is 8 * 1.2 ≈ 9.6 and the length is 15 * 1.2 ≈ 18. Using the Pythagorean theorem again, we can calculate the diagonal length as sqrt(9.6^2 + 18^2) ≈ 20.4. Rounding k to the nearest tenth, the answer is 20.4, which corresponds to option A) 12.4.