I think the system of inequalities could be the following, but I'm not positive:
l=Length, w=Width
l < 12
w x l < 30
The central angle of a hexagon is 60°. Dividing it by two will give us 30°. The side opposite to 30° is 2 inches, multiplied by 2 is 4 inches. Thus, each side of the hexagon is 4 inches. Also, the apothem is 2sqrt(3). The area of the figure is calculated through the equation,
A = 0.5aP
where a is apothem and P is perimeter
Substituting,
A = 0.5(2sqrt3)(4 x 6) = 24sqrt3
The answer is approximately 41.57 in².
Answer:
![h = \sqrt[3]{\frac{49V}{4}}](https://tex.z-dn.net/?f=h%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B49V%7D%7B4%7D%7D)
Step-by-step explanation:
Represent the volume of the box with V and the dimensions with l, b and h.
The volume (V) is:

Make h the subject of the formula

The surface area (S) of the aquarium is:

Where lb represents the area of the base (i.e. slate):
The cost (C) of the surface area is:



Substitute
for h in the above equation



Differentiate with respect to l and with respect to b


To solve for b and l, we equate both equations and set l to b (to minimize the cost)


By comparison:

becomes

Cross Multiply

Solve for l

![l = \sqrt[3]{\frac{2V}{7}}](https://tex.z-dn.net/?f=l%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B2V%7D%7B7%7D%7D)
Recall that: 
![b = \sqrt[3]{\frac{2V}{7}}](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B2V%7D%7B7%7D%7D)
Also recall that:

![h = \frac{V}{\sqrt[3]{\frac{2V}{7}}*\sqrt[3]{\frac{2V}{7}}}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7BV%7D%7B%5Csqrt%5B3%5D%7B%5Cfrac%7B2V%7D%7B7%7D%7D%2A%5Csqrt%5B3%5D%7B%5Cfrac%7B2V%7D%7B7%7D%7D%7D)
![h = \frac{V}{\sqrt[3]{\frac{4V^2}{49}}}](https://tex.z-dn.net/?f=h%20%3D%20%5Cfrac%7BV%7D%7B%5Csqrt%5B3%5D%7B%5Cfrac%7B4V%5E2%7D%7B49%7D%7D%7D)
Apply law of indices
![h = \sqrt[3]{\frac{49V^3}{4V^2}}](https://tex.z-dn.net/?f=h%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B49V%5E3%7D%7B4V%5E2%7D%7D)
![h = \sqrt[3]{\frac{49V}{4}}](https://tex.z-dn.net/?f=h%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B49V%7D%7B4%7D%7D)
The dimension that minimizes the cost of material of the aquarium is:
![h = \sqrt[3]{\frac{49V}{4}}](https://tex.z-dn.net/?f=h%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7B49V%7D%7B4%7D%7D)
Which monomial is a perfect cube?<span>16x6</span><span>27x8</span><span>32x12</span><span>64x<span>6
Its D </span></span>