Question

Help! Please? (Brainlyist and 15pts)

There are 7 similarly shaped figures in a set. The largest figure is 7 inches tall. Each of the other figures is 1 inch shorter than the next larger figure. The surface area of the largest figure is 588 in^2 . What is the surface area of the smallest figure?

Answer 1

588/7=84

7x84=588

1x84=84 in^2  smallest figure

Answer 2

Figure 1 (tallest): height (h) = 7
Figure 2: h = 7 - 1 = 6
Figure 3: h = 6 - 1 = 5
Figure 4: h = 5 - 1 = 4
Figure 5: h = 4 - 1 = 3
Figure 6: h = 3 - 1 = 2
Figure 7 (shortest): h = 2 - 1 = 1

Let's use a rectangular prism to demonstrate how height affects surface area.
(I am making the assumption that the length and width are the same for all seven figures.)
S.A.(tallest)= 2(L x w) + 2(w x h) + 2(L x h)
           588 = 2(L x w) + 2(w x 7) + 2(L x 7)
           588 =    2Lw    +    14w     -     14L

Solve for one of the variables (I am choosing to solve for w):
            588 - 14L = 2Lw + 14w
            588 - 14L = 2w(2L + 7)
            $\frac{588 - 14L}{2(2L + 7)} = w$
            $\frac{294 - 7L}{2L + 7} = w$
Now, plug w into the equation for the surface area of the shortest figure:
reminder that the height of the shortest figure is 1 in (h = 1)
S.A. (shortest) = 2(L x w) + 2(w x h) + 2(L x h)
                        = 2(L x w) + 2(w) + 2(L)
                        = 2L · $\frac{294 - 7L}{2L + 7}$ + 2 · $\frac{294 - 7L}{2L + 7}$ + 2L
                        =  $\frac{588L - 14L^{2} + 588L -14L + 4L^{2} + 14L}{2L + 7}$
                        = $\frac{-10L^{2} + 1176}{2L + 7}$



I cannot go any further without knowing the length.
Hope this is what you are looking for!