answer:First, we need to find out the values x where g(x) = 8. The equation becomes: [ x^3 - 3x = 8 ] [ x^3 - 3x - 8 = 0 ] Using the rational root theorem or synthetic division, we find that x = 2 is a root. Factoring out (x-2) from x^3 - 3x - 8: [ x^3 - 3x - 8 = (x-2)(x^2 + 2x + 4) ] The quadratic x^2 + 2x + 4 has no real roots (since the discriminant b^2 - 4ac = 4 - 16 = -12 < 0). Thus, the only real solution to g(x) = 8 is x = 2. Next, we need to find where g(x) = 2. This leads to: [ x^3 - 3x = 2 ] [ x^3 - 3x - 2 = 0 ] Again, x = 2 is a root. Factoring out (x-2): [ x^3 - 3x - 2 = (x-2)(x^2 + 2x + 1) = (x-2)(x+1)^2 ] Here, x = 2 and x = -1 (double root) are solutions. Continuing, we find where g(x) = -1 for the above solutions. It leads to: [ x^3 - 3x = -1 ] [ x^3 - 3x + 1 = 0 ] None of x = 2 or x = -1 satisfy this. Checking for other roots using synthetic division or trial doesn't give any easy real solutions. Thus, the layers simplify as: - g(g(g(g(d)))) = 8 directly implies g(g(g(d))) = 2, - g(g(g(d))) = 2 implies g(g(d)) = 2, - g(g(d)) = 2 implies g(d) = 2, and - g(d) = 2 implies d = 2 or d = -1 (double root). Counting distinct values, d = 2 and d = -1. There are boxed{2} distinct real numbers d that satisfy the equation.

answer:This problem involves calculating the volume of a rectangular prism and applying derivative knowledge. It is a moderate difficulty problem. Let the three dimensions of the rectangular prism be a, b, and c. Then, we have: a + b + c = 9, and ab + bc + ac = 24. Simplifying the volume V = abc, we get: V = c(c^2 - 9c + 24). The range of c is [1, 5]. Differentiating with respect to c, we obtain: V' = 3(c-2)(c-4). Thus, the function is increasing on (0,2) and (4,9), and decreasing on (2,4). When c=2, V=20, and when c=4, V=16. Therefore, the range of possible values for the volume of the rectangular prism is boxed{[16, 20]}.

answer:First, add the terms based on their powers: - **Cubic terms (d^3):** 5d^3 + 3d^3 = 8d^3 - **Quadratic terms (d^2):** 18d^2 - **Linear terms (d):** 16d + 4d = 20d - **Constant terms:** 15 + 3 = 18 Now combine these results: [ 20d + 18 + 18d^2 + 8d^3 ] Thus, we have a = 20, b = 18, c = 18, and e = 8. Therefore, a+b+c+e = 20 + 18 + 18 + 8 = boxed{64}.

answer:The stone pillar is in the shape of a right circular cylinder, and it must fit into the crate so that it rests upright when the crate sits on at least one of its six sides. The radius of the pillar is 7 feet, so the diameter is 14 feet. This means that the smallest side of the crate must be at least 14 feet to accommodate the diameter of the pillar when the crate is resting on that side. Since the crate measures some feet by 8 feet by 12 feet on the inside, and we know that the smallest side must be at least 14 feet to fit the pillar, we can conclude that the smallest side of the crate is the 8-foot side. The pillar will fit into the crate when it rests on the side that is 12 feet by some feet, with the 12-foot side being the height of the crate in that orientation. Therefore, the length of the crate's smallest side is boxed{8} feet.