Answer:
cm
Step-by-step explanation:
The volume of the box is:
V = height * length * width
V = x*(66 - 2*x)*(90 - 2*x)
V = (66*x - 2*x^2)*(90 - 2*x)
V = 5940*x - 132*x^2 - 180*x^2 + 4*x^3
V = 4*x^3 - 312*x^2 + 5940*x
where x is the length of the sides of the squares, in cm.
The mathematical problem is :
Maximize: V = 4*x^3 - 312*x^2 + 5940*x
subject to:
x > 0
2*x < 66 <=> x < 33
In the maximum, the first derivative of V, dV/dx, is equal to zero
dV/dx = 12*x^2 - 624*x + 5940
From quadratic formula
![x = \frac{-b \pm \sqrt{b^2 - 4(a)(c)}}{2(a)}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E2%20-%204%28a%29%28c%29%7D%7D%7B2%28a%29%7D%20)
![x = \frac{624 \pm \sqrt{(-624)^2 - 4(12)(5940)}}{2(12)}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B624%20%5Cpm%20%5Csqrt%7B%28-624%29%5E2%20-%204%2812%29%285940%29%7D%7D%7B2%2812%29%7D%20)
![x = \frac{624 \pm \sqrt{104256}}{24}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B624%20%5Cpm%20%5Csqrt%7B104256%7D%7D%7B24%7D%20)
![x = \frac{624 \pm \sqrt{2^6*3^2*181}}{24}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B624%20%5Cpm%20%5Csqrt%7B2%5E6%2A3%5E2%2A181%7D%7D%7B24%7D%20)
![x = \frac{624 \pm 8*3*\sqrt{181}}{24}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B624%20%5Cpm%208%2A3%2A%5Csqrt%7B181%7D%7D%7B24%7D%20)
![x_1 = \frac{624 + 24*\sqrt{181}}{24}](https://tex.z-dn.net/?f=x_1%20%3D%20%5Cfrac%7B624%20%2B%2024%2A%5Csqrt%7B181%7D%7D%7B24%7D%20)
![x_1 = 26 + \sqrt{181}](https://tex.z-dn.net/?f=x_1%20%3D%2026%20%2B%20%5Csqrt%7B181%7D)
![x_2 = \frac{624 - 24*\sqrt{181}}{24}](https://tex.z-dn.net/?f=x_2%20%3D%20%5Cfrac%7B624%20-%2024%2A%5Csqrt%7B181%7D%7D%7B24%7D%20)
![x_2 = 26 - \sqrt{181}](https://tex.z-dn.net/?f=x_2%20%3D%2026%20-%20%5Csqrt%7B181%7D)
But
, then is not the correct answer.