Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the indicated functions. Express the area \(A\) of a square as a function of its diagonal \(d\); express the diagonal \(d\) of a square as a function of its area \(A\)

Step-by-Step Solution

Answer

The area as a function of diagonal is \(A(d) = \frac{d^2}{2}\), and the diagonal as a function of area is \(d(A) = \sqrt{2A}\).

Step 1: Understanding the Problem

We need to express the area of a square as a function of its diagonal and vice versa. First, we identify the relationship between the side and diagonal of a square. Given a square with side length \(s\), its diagonal \(d\) can be expressed using the Pythagorean theorem as \(d = s\sqrt{2}\).

Step 2: Express Area as a Function of Diagonal

Start with the diagonal formula \(d = s\sqrt{2}\) and solve for \(s\): \[ s = \frac{d}{\sqrt{2}} \] The area \(A\) of the square as a function of the side \(s\) is \(A = s^2\). Substitute \(s = \frac{d}{\sqrt{2}}\) into this equation:\[ A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2} \] Thus, the area \(A\) as a function of the diagonal \(d\) is \(A(d) = \frac{d^2}{2}\).

Step 3: Express Diagonal as a Function of Area

To find \(d\) as a function of \(A\), start from the equation \(A = \frac{d^2}{2}\). Solve for \(d\): \[ d^2 = 2A \] \[ d = \sqrt{2A} \] Thus, the diagonal \(d\) as a function of the area \(A\) is \(d(A) = \sqrt{2A}\).

Key Concepts

Diagonal of a Square

The diagonal of a square is a line that connects opposite corners of the square. It splits the square into two congruent right-angled triangles. Each triangle has sides that are the same length as the sides of the square. The diagonal can be thought of as the hypotenuse of these triangles.

To find the formula for the diagonal of a square, we use the relationship between the side length and the diagonal. If the side of the square is denoted as \( s \), the diagonal \( d \) can be found using:

• The Pythagorean theorem
• Expressed as \( d = s\sqrt{2} \)

This explains how the diagonal is always \( \sqrt{2} \) times the length of the side. This special factor comes from the square's property of a 45-degree right triangle where the hypotenuse is \( \sqrt{2} \) longer than the legs.

Area Function

The area function of a square expresses how the area is calculated based on the dimensions of the square.

For any square, its area \( A \) is the square of its side length. So, if the length of one side of a square is \( s \), the area can be expressed as:

• \( A = s^2 \)

When we relate the area to the diagonal, recall that from the diagonal formula, \( s = \frac{d}{\sqrt{2}} \).

Plug this into the area formula to see how the area depends on the diagonal:

• \( A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2} \)

The area function in terms of the diagonal illustrates how the diagonal inherently affects the amount of space inside the square.

Pythagorean Theorem

The Pythagorean Theorem is a vital concept when studying squares, especially when considering their diagonals. Named after the ancient Greek mathematician Pythagoras, this theorem defines the relationship between the sides of a right triangle.

A basic formulation of the Pythagorean Theorem states:

• For a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \): \( a^2 + b^2 = c^2 \)

In a square, the diagonal acts as the hypotenuse of the triangle formed by its sides.

Using this, if each side of the square is \( s \):

• The legs are both \( s \)
• The hypotenuse, or diagonal \( d \), follows from \( s^2 + s^2 = d^2 \)

Simplifying gives \( d = s\sqrt{2} \). The theorem ensures precision in finding the diagonal and showcases its dependency on square properties.