21. shipping crates a square - based, box - shaped shipping crate is designed to have a volume of 16 ft³…

Explanation:

Step1: Set up variables and volume formula

Let the side length of the square base be (x) (in feet) and the height be (h) (in feet). The volume (V=x^{2}h), and since (V = 16), we have (h=\frac{16}{x^{2}}).

Let the cost per square - foot of the side material be (c). Then the cost per square - foot of the base material is (2c) and the cost per square - foot of the top material is (\frac{c}{2}).

The surface area of the base is (A_{base}=x^{2}), the surface area of the top is (A_{top}=x^{2}), and the surface area of the four sides is (A_{sides}=4xh).

The cost function (C) is: [ \begin{align*} C&=2cA_{base}+cA_{sides}+\frac{c}{2}A_{top}\ &=2cx^{2}+4cxh+\frac{c}{2}x^{2}\ &=\left(2c +\frac{c}{2}\right)x^{2}+4cxh\ &=\frac{5c}{2}x^{2}+4cxh \end{align*} ]

Substitute (h = \frac{16}{x^{2}}) into the cost function: [ C(x)=\frac{5c}{2}x^{2}+4cx\cdot\frac{16}{x^{2}}=\frac{5c}{2}x^{2}+\frac{64c}{x} ]

Step2: Find the derivative of the cost function

Differentiate (C(x)) with respect to (x). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have: [ C^\prime(x)=5cx-\frac{64c}{x^{2}} ]

Set (C^\prime(x) = 0) (since at the minimum of a function (y = f(x)), (f^\prime(x)=0)): [ \begin{align*} 5cx-\frac{64c}{x^{2}}&=0\ 5cx&=\frac{64c}{x^{2}}\ 5x^{3}&=64\ x^{3}&=\frac{64}{5}\ x&=\sqrt[3]{\frac{64}{5}}=\frac{4}{\sqrt[3]{5}}\approx2.34 \end{align*} ]

Step3: Find the second - derivative and confirm it's a minimum

Differentiate (C^\prime(x)) with respect to (x) to get the second - derivative (C^{\prime\prime}(x)): [ C^{\prime\prime}(x)=5c+\frac{128c}{x^{3}} ]

When (x=\frac{4}{\sqrt[3]{5}}), (C^{\prime\prime}(x)>0) (since (c>0) and (x>0)), so (x=\frac{4}{\sqrt[3]{5}}) gives a minimum of the cost function.

Then (h=\frac{16}{x^{2}}), substitute (x = \frac{4}{\sqrt[3]{5}}) into (h): [ h=\frac{16}{\left(\frac{4}{\sqrt[3]{5}}\right)^{2}}=\frac{16}{\frac{16}{\sqrt[3]{25}}}=\sqrt[3]{25}\approx2.92 ]

Answer:

The side length of the square base (x=\sqrt[3]{\frac{64}{5}}\approx2.34) ft and the height (h = \sqrt[3]{25}\approx2.92) ft.