20. an open - top rectangular box has a volume of 250 cm³. the width of the box is 5 cm. the cost of the box…

Explanation:

Step1: Find the length - height relationship

Let the length be $l$ and height be $h$. Given volume $V = l\times5\times h=250$, so $l=\frac{250}{5h}=\frac{50}{h}$.

Step2: Calculate the cost function

The area of the base is $A_{base}=l\times5 = 5l$, and the cost of the base is $2\times5l = 10l$. The area of the two sides with dimensions $5\times h$ is $2\times5h = 10h$, and the area of the two sides with dimensions $l\times h$ is $2lh$. The total cost function $C$ is $C = 10l+10h + 2lh$. Substitute $l=\frac{50}{h}$ into the cost - function: [ \begin{align*} C&=10\times\frac{50}{h}+10h+2\times\frac{50}{h}\times h\ &=\frac{500}{h}+10h + 100 \end{align*} ]

Step3: Differentiate the cost function

Differentiate $C(h)$ with respect to $h$. $C^\prime(h)=-\frac{500}{h^{2}}+10$.

Step4: Find the critical points

Set $C^\prime(h) = 0$. Then $-\frac{500}{h^{2}}+10 = 0$. [ \begin{align*} \frac{500}{h^{2}}&=10\ h^{2}& = 50\ h&=\sqrt{50}=5\sqrt{2}\text{ cm} \end{align*} ]

Step5: Check the second - derivative

Differentiate $C^\prime(h)$ to get the second - derivative $C^{\prime\prime}(h)=\frac{1000}{h^{3}}$. When $h = 5\sqrt{2}$, $C^{\prime\prime}(5\sqrt{2})=\frac{1000}{(5\sqrt{2})^{3}}=\frac{1000}{250\sqrt{8}}>0$, so the cost function has a minimum at $h = 5\sqrt{2}$.

Step6: Calculate the minimum cost

When $h = 5\sqrt{2}$, $l=\frac{50}{5\sqrt{2}} = 5\sqrt{2}$. [ \begin{align*} C&=\frac{500}{5\sqrt{2}}+10\times5\sqrt{2}+100\ &=\frac{100}{\sqrt{2}}+50\sqrt{2}+100\ &=50\sqrt{2}+50\sqrt{2}+100\ &=100 + 100\sqrt{2}\approx100+100\times1.414 = 241.4 \end{align*} ]

Answer:

$100 + 100\sqrt{2}\approx241.4$ dollars