1.1 Q-3

Question Statement

Express the following:

a. The perimeter $P$ of a square as a function of its area $A$ .

b. The area $A$ of a circle as a function of its circumference $C$ .

c. The volume $V$ of a cube as a function of the area $A$ of its base.

Background and Explanation

In these problems, we are asked to express one geometric property (such as perimeter, area, or volume) in terms of another. This involves applying algebraic manipulations to known geometric formulas, such as the formula for the perimeter of a square, the area of a circle, and the volume of a cube.

Solution

a. The perimeter $P$ of a square as a function of its area $A$

Let the side of the square be denoted by $x$ .

The perimeter $P$ of a square is given by:

P = 4x

The area $A$ of the square is given by:

A = x^2

From the area formula, solve for $x$ :

x = \frac{P}{4}

Substitute $x$ into the area formula:

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

Rearranging the equation, we get:

16A = P^2 \quad \Rightarrow \quad P = 4\sqrt{A}

b. The area $A$ of a circle as a function of its circumference $C$

Let the radius of the circle be $r$ .

The circumference $C$ of a circle is given by:

C = 2\pi r

The area $A$ of the circle is given by:

A = \pi r^2

From the circumference formula, solve for $r$ :

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

Substitute this expression for $r$ into the area formula:

A = \pi \left( \frac{C}{2\pi} \right)^2 = \frac{C^2}{4\pi^2}

Therefore, we have:

4\pi A = C^2 \quad \Rightarrow \quad A = \frac{C^2}{4\pi}

c. The volume $V$ of a cube as a function of the area $A$ of its base

Let the side length of the cube be $x$ .

The area of the base of the cube is:

A = x^2

The volume $V$ of the cube is:

V = x^3

From the area formula, solve for $x$ :

x = \sqrt{A}

Substitute this expression for $x$ into the volume formula:

V = (\sqrt{A})^3 = A^{3/2}

Key Formulas or Methods Used

Square Perimeter as a Function of Area:

P = 4\sqrt{A}

Circle Area as a Function of Circumference:

A = \frac{C^2}{4\pi}

Cube Volume as a Function of Base Area:

V = A^{3/2}

Summary of Steps

For the square, solve $x = \frac{P}{4}$ and substitute into the area formula to find $P = 4\sqrt{A}$ .
For the circle, solve $r = \frac{C}{2\pi}$ and substitute into the area formula to find $A = \frac{C^2}{4\pi}$ .
For the cube, solve $x = \sqrt{A}$ and substitute into the volume formula to find $V = A^{3/2}$ .