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Optimization

AP Calculus AB· difficulty 3/5

A box with square base and <strong>open</strong> top has volume 108 m3108~\text{m}^3. The dimensions minimizing surface area are

  • A

    Base 99, height 43\tfrac{4}{3}

  • B

    Base 44, height 274\tfrac{27}{4}

  • C

    Base 66, height 33

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  • D

    Cube of side 1083\sqrt[3]{108}

Explanation

S=s2+4shS = s^2 + 4 s h with s2h=108s^2 h = 108, so S(s)=s2+432/sS(s) = s^2 + 432/s. S′=2s−432/s2=0⇒s3=216⇒s=6,h=3S' = 2s - 432/s^2 = 0 \Rightarrow s^3 = 216 \Rightarrow s = 6, h = 3.

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