a microwaveable cup - of - soup package needs to be constructed in the shape of cylinder to hold 600 cubic…

Explanation:

Step1: Express height in terms of radius

Given $V=\pi r^{2}h = 600$, so $h=\frac{600}{\pi r^{2}}$.

Step2: Write the cost - function

The cost of the bottom and sides is $0.02$ cents per square centimeter and the cost of the top is $0.05$ cents per square centimeter. The cost function $C(r)$ is $C(r)=0.02(2\pi r h+\pi r^{2})+0.05\pi r^{2}$. Substitute $h = \frac{600}{\pi r^{2}}$ into the cost - function: [ \begin{align*} C(r)&=0.02\left(2\pi r\times\frac{600}{\pi r^{2}}+\pi r^{2}\right)+0.05\pi r^{2}\ &=0.02\left(\frac{1200}{r}+\pi r^{2}\right)+0.05\pi r^{2}\ &=\frac{24}{r}+0.02\pi r^{2}+0.05\pi r^{2}\ &=\frac{24}{r}+0.07\pi r^{2} \end{align*} ]

Step3: Find the derivative of the cost - function

Differentiate $C(r)$ with respect to $r$: $C^\prime(r)=-\frac{24}{r^{2}} + 0.14\pi r$.

Step4: Set the derivative equal to zero and solve for $r$

[ \begin{align*} -\frac{24}{r^{2}}+0.14\pi r&=0\ 0.14\pi r&=\frac{24}{r^{2}}\ 0.14\pi r^{3}&=24\ r^{3}&=\frac{24}{0.14\pi}\ r^{3}&=\frac{120}{0.7\pi}\ r&\approx\sqrt[3]{\frac{120}{0.7\pi}}\approx\sqrt[3]{\frac{120}{2.1991}}\approx\sqrt[3]{54.56}\approx3.79 \end{align*} ]

Step5: Find the height

Substitute $r\approx3.79$ into $h=\frac{600}{\pi r^{2}}$: $h=\frac{600}{\pi\times(3.79)^{2}}\approx\frac{600}{45.16}\approx13.28$.

Step6: Find the minimum cost

Substitute $r\approx3.79$ into the cost - function $C(r)$: [ \begin{align*} C(3.79)&=\frac{24}{3.79}+0.07\pi\times(3.79)^{2}\ &\approx6.33 + 0.07\times3.14\times14.36\ &\approx6.33+3.14\ &\approx9.47 \end{align*} ]

Answer:

Radius: $3.79$ cm Height: $13.28$ cm Minimum cost: $9.47$ cents