Answer:
Height = 7.5 inches
Width = 10 inches
Length = 30 inches
Step-by-step explanation:
Volume of the shape (V) = length (l) * width (w) * height (h)
From the question, the length = 3w
So, V = 3w * w * h
V = 3w²h
h = V/3w²
h = 2250/3w²
Therefore h=V/(3w^2) where V is the volume.
Considering the fact that the top of the rectangular box is missing
The Surface Area (S) = (l*w) + 2(l*h) + 2(w*h) --------------------- (replace l with 3w and h with 2250/3w²)
S = 3w*w + 2(3w)(2250/3w²) + 2w(2250/3w²)
S = 3w² + 4500/w² + 1500/w
S = 3w² + 6000/w
We need to find the minimum of the above function (equation) by finding its first derivative with respect to A
Differentiating the above function, we have
0 = 6w - 6000/w²
6000/w² = 6w
6000 = 6w³ ------------ Divide through by 6
1000 = w³ ---------------- Find the cube root of both sides
10 = w
w = 10
Hence, width of the rectangular box is 10 inches
To solve height, we use
h = 2250/3w²
So, h = 2250/(3*10²)
h = 2250/300
h = 7.5 inches
To find the length, we use
l = 3w
So, l = 3 * 10
l = 30 inches
