Answer:
![C(x)=\dfrac{20x^3+1715000}{x}\\$Minimum cost, C(35)=\$29,400](https://tex.z-dn.net/?f=C%28x%29%3D%5Cdfrac%7B20x%5E3%2B1715000%7D%7Bx%7D%5C%5C%24Minimum%20cost%2C%20C%2835%29%3D%5C%2429%2C400)
The dimensions that will lead to minimum cost of the box are a base length of 35 cm and a height of 140 cm.
Step-by-step explanation:
Volume of the Square-Based box=171,500 cubic cm
Let the length of a side of the base=x cm
Volume ![=x^2h](https://tex.z-dn.net/?f=%3Dx%5E2h)
![x^2h=171,500\\h=\dfrac{171500}{x^2}](https://tex.z-dn.net/?f=x%5E2h%3D171%2C500%5C%5Ch%3D%5Cdfrac%7B171500%7D%7Bx%5E2%7D)
The material for the top and bottom of the box costs $10.00 per square centimeter.
Surface Area of the Top and Bottom ![=2x^2](https://tex.z-dn.net/?f=%3D2x%5E2)
Therefore, Cost of the Top and Bottom ![=\$10X2x^2=20x^2](https://tex.z-dn.net/?f=%3D%5C%2410X2x%5E2%3D20x%5E2)
The material for the sides costs $2.50 per square centimeter.
Surface Area of the Sides=4xh
Cost of the sides=$2.50 X 4xh =10xh
![\text{Substitute h}$=\dfrac{171500}{x^2} $into 10xh\\Cost of the sides=10x(\dfrac{171500}{x^2})=\dfrac{1715000}{x}](https://tex.z-dn.net/?f=%5Ctext%7BSubstitute%20h%7D%24%3D%5Cdfrac%7B171500%7D%7Bx%5E2%7D%20%24into%2010xh%5C%5CCost%20of%20the%20sides%3D10x%28%5Cdfrac%7B171500%7D%7Bx%5E2%7D%29%3D%5Cdfrac%7B1715000%7D%7Bx%7D)
Therefore, total Cost of the box
![= 20x^2+\dfrac{1715000}{x}\\C(x)=\dfrac{20x^3+1715000}{x}](https://tex.z-dn.net/?f=%3D%2020x%5E2%2B%5Cdfrac%7B1715000%7D%7Bx%7D%5C%5CC%28x%29%3D%5Cdfrac%7B20x%5E3%2B1715000%7D%7Bx%7D)
To find the minimum total cost, we solve for the critical points of C(x). This is obtained by equating its derivative to zero and solving for x.
![C'(x)=\dfrac{40x^3-1715000}{x^2}\\\dfrac{40x^3-1715000}{x^2}=0\\40x^3-1715000=0\\40x^3=1715000\\x^3=1715000\div 40\\x^3=42875\\x=\sqrt[3]{42875}=35](https://tex.z-dn.net/?f=C%27%28x%29%3D%5Cdfrac%7B40x%5E3-1715000%7D%7Bx%5E2%7D%5C%5C%5Cdfrac%7B40x%5E3-1715000%7D%7Bx%5E2%7D%3D0%5C%5C40x%5E3-1715000%3D0%5C%5C40x%5E3%3D1715000%5C%5Cx%5E3%3D1715000%5Cdiv%2040%5C%5Cx%5E3%3D42875%5C%5Cx%3D%5Csqrt%5B3%5D%7B42875%7D%3D35)
Recall that:
![h=\dfrac{171500}{x^2}\\Therefore:\\h=\dfrac{171500}{35^2}=140cm](https://tex.z-dn.net/?f=h%3D%5Cdfrac%7B171500%7D%7Bx%5E2%7D%5C%5CTherefore%3A%5C%5Ch%3D%5Cdfrac%7B171500%7D%7B35%5E2%7D%3D140cm)
The dimensions that will lead to minimum costs are base length of 35cm and height of 140cm.
Therefore, the minimum total cost, at x=35cm
![C(35)=\dfrac{20(35)^3+1715000}{35}=\$29,400](https://tex.z-dn.net/?f=C%2835%29%3D%5Cdfrac%7B20%2835%29%5E3%2B1715000%7D%7B35%7D%3D%5C%2429%2C400)