Question

part c — stem challenge question
you are designing a package that uses the least material but holds at least 500 cm³.
· which shape would you choose: cylinder, rectangular prism, or sphere?
· explain your choice mathematically with formulas and reasoning.
· sketch your design below.

Explanation:

Step1: Volume formulas

Volume of cylinder \(V=\pi r^{2}h\), volume of rectangular prism \(V = l\times w\times h\), volume of sphere \(V=\frac{4}{3}\pi r^{3}\).

Step2: Surface - area formulas

Surface area of cylinder \(S = 2\pi r(r + h)\), surface area of rectangular prism \(S=2(lw+lh + wh)\), surface area of sphere \(S = 4\pi r^{2}\).

Step3: Comparison for given volume

For a given volume, the sphere has the minimum surface - area among these three shapes. Mathematically, if \(V=\frac{4}{3}\pi r^{3}\geq500\), then \(r\geq\sqrt[3]{\frac{3\times500}{4\pi}}\approx3.4\) cm. And \(S = 4\pi r^{2}\). For a cylinder with \(V=\pi r^{2}h = 500\), \(h=\frac{500}{\pi r^{2}}\), \(S=2\pi r(r+\frac{500}{\pi r^{2}})=2\pi r^{2}+\frac{1000}{r}\). For a rectangular prism, assume \(l = w=h\) (a cube for simplicity, since for a rectangular prism, the cube has the minimum surface - area among rectangular prisms for a given volume), \(V=a^{3}=500\), \(a=\sqrt[3]{500}\approx7.9\) cm, \(S = 6a^{2}\).

Answer:

Sphere. Because for a given volume, the sphere has the minimum surface - area among cylinder, rectangular prism and sphere.