First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightFunctions And Graphschevron_rightExercise 2.1

Solution

(a) Area of a square as a function of its perimeter $P$

Let side length be $s$ .

P=4s\Rightarrow s=\frac{P}{4}

A=s^2=\left(\frac{P}{4}\right)^2=\frac{P^2}{16}

\boxed{A(P)=\frac{P^2}{16}}

(b) Circumference $C$ as a function of area $A$

Let radius be $r$ .

A=\pi r^2\Rightarrow r=\sqrt{\frac{A}{\pi}}

C=2\pi r=2\pi\sqrt{\frac{A}{\pi}}=2\sqrt{\pi A}

\boxed{C(A)=2\sqrt{\pi A}}

(c) Surface area $S$ of a cube as a function of volume $V$

Let side length be $s$ .

V=s^3\Rightarrow s=\sqrt[3]{V}

S=6s^2=6\left(\sqrt[3]{V}\right)^2=6V^{2/3}

\boxed{S(V)=6V^{2/3}}

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