Step-by-step explanation:
V = 1.5 × 2 × 1 = 3 m³ = 3000 L
tiempo = 3000 ÷ 150 = 20 minutos
Given:
The function is
![f(x)=(x-5)^2(3-x)^2](https://tex.z-dn.net/?f=f%28x%29%3D%28x-5%29%5E2%283-x%29%5E2)
To find:
The derivative of the given function.
Solution:
Chain rule of differentiation:
![[f(g(x))]'=f'(g(x))g'(x)](https://tex.z-dn.net/?f=%5Bf%28g%28x%29%29%5D%27%3Df%27%28g%28x%29%29g%27%28x%29)
Product rule of differentiation:
![[f(x)g(x)]'=f(x)g'(x)+g(x)f'(x)](https://tex.z-dn.net/?f=%5Bf%28x%29g%28x%29%5D%27%3Df%28x%29g%27%28x%29%2Bg%28x%29f%27%28x%29)
We have,
![f(x)=(x-5)^2(3-x)^2](https://tex.z-dn.net/?f=f%28x%29%3D%28x-5%29%5E2%283-x%29%5E2)
Differentiate with respect to x.
![f'(x)=(x-5)^2\dfrac{d}{dx}(3-x)^2+(3-x)^2\dfrac{d}{dx}(x-5)^2](https://tex.z-dn.net/?f=f%27%28x%29%3D%28x-5%29%5E2%5Cdfrac%7Bd%7D%7Bdx%7D%283-x%29%5E2%2B%283-x%29%5E2%5Cdfrac%7Bd%7D%7Bdx%7D%28x-5%29%5E2)
![f'(x)=(x-5)^2[2(3-x)(0-1)]+(3-x)^2[2(x-5)(1-0)]](https://tex.z-dn.net/?f=f%27%28x%29%3D%28x-5%29%5E2%5B2%283-x%29%280-1%29%5D%2B%283-x%29%5E2%5B2%28x-5%29%281-0%29%5D)
![f'(x)=(x^2-10x+25)(-6+2x)+(9-6x+x^2)(2x-10)](https://tex.z-dn.net/?f=f%27%28x%29%3D%28x%5E2-10x%2B25%29%28-6%2B2x%29%2B%289-6x%2Bx%5E2%29%282x-10%29)
![f'(x)=(x^2)(-6)+(-10x)(-6)+(25)(-6)+(x^2)(2x)-10x(2x)+25(2x)+(9)(2x)+(-6x)(2x)+x^2(2x)+9(-10)+(-6x)(-10)+x^2(-10)](https://tex.z-dn.net/?f=f%27%28x%29%3D%28x%5E2%29%28-6%29%2B%28-10x%29%28-6%29%2B%2825%29%28-6%29%2B%28x%5E2%29%282x%29-10x%282x%29%2B25%282x%29%2B%289%29%282x%29%2B%28-6x%29%282x%29%2Bx%5E2%282x%29%2B9%28-10%29%2B%28-6x%29%28-10%29%2Bx%5E2%28-10%29)
On further simplification, we get
![f'(x)=-6x^2+60x-150+2x^3-20x^2+50x+18x-12x^2+2x^3-90+60x-10x^2](https://tex.z-dn.net/?f=f%27%28x%29%3D-6x%5E2%2B60x-150%2B2x%5E3-20x%5E2%2B50x%2B18x-12x%5E2%2B2x%5E3-90%2B60x-10x%5E2)
![f'(x)=(2x^3+2x^3)+(-6x^2-20x^2-12x^2-10x^2)+(60x+50x+18x+60x)+(-90-150)](https://tex.z-dn.net/?f=f%27%28x%29%3D%282x%5E3%2B2x%5E3%29%2B%28-6x%5E2-20x%5E2-12x%5E2-10x%5E2%29%2B%2860x%2B50x%2B18x%2B60x%29%2B%28-90-150%29)
![f'(x)=4x^3-48x^2+188x-240](https://tex.z-dn.net/?f=f%27%28x%29%3D4x%5E3-48x%5E2%2B188x-240)
Therefore, the derivative of the given function is
.
We have to find the volume of tank. The tank is in cuboid or rectangular prism shape
Volume = depth*height*width
= 12*24*18
= 5184 cubic inches