Question

question 9 · 1 point you have been asked to design a can with a volume of 672 cm³ that is shaped like a right - circular cylinder. the can will have a closed top. what radius r and height h, in centimeters, would minimize the amount of material needed to construct this can? enter an exact answer. provide your answer below: r = cm h = cm

Explanation:

Step1: Recall volume and surface - area formulas

The volume formula for a right - circular cylinder is $V=\pi r^{2}h$, and we know $V = 672$, so $\pi r^{2}h=672$, which gives $h=\frac{672}{\pi r^{2}}$. The surface - area formula for a closed - top cylinder is $A = 2\pi r^{2}+2\pi rh$.

Step2: Substitute $h$ into the surface - area formula

Substitute $h=\frac{672}{\pi r^{2}}$ into $A$:

$$
LATEXBLOCK0
$$

Step3: Find the derivative of $A(r)$

Using the power rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$.

$$
LATEXBLOCK1
$$

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

$$
LATEXBLOCK2
$$

Step5: Find the value of $h$

Substitute $r = \sqrt[3]{\frac{336}{\pi}}$ into $h=\frac{672}{\pi r^{2}}$:

$$
LATEXBLOCK3
$$

Answer:

$r=\sqrt[3]{\frac{336}{\pi}}\text{ cm}$, $h = 2\sqrt[3]{\frac{336}{\pi}}\text{ cm}$