Problem 58 - Calculus Single Variable

Question

What is the rate of change of the area of an equilateral triangle with respect to its side length when that side length is 8 inches?

Step-by-Step Solution

Answer

The rate of change is \( 4\sqrt{3} \) square inches per inch.

Step 1: Find the formula for the area of an equilateral triangle

The area of an equilateral triangle with side length \( s \) is given by the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \]. This formula will help us calculate the area based on a given side length.

Step 2: Differentiate with respect to the side length

To find the rate of change of the area with respect to the side length \( s \), differentiate the area formula with respect to \( s \). The derivative is \[ \frac{dA}{ds} = \frac{\sqrt{3}}{4} \cdot 2s = \frac{\sqrt{3}}{2}s \]. This gives us the rate of change as a function of \( s \).

Step 3: Evaluate the derivative at \( s = 8 \) inches

Substitute \( s = 8 \) into the derivative: \[ \frac{dA}{ds} = \frac{\sqrt{3}}{2} \cdot 8 = 4\sqrt{3} \]. This result is the rate of change of the area when the side length is 8 inches.

Key Concepts

Equilateral Triangle

An equilateral triangle is a type of polygon where all three sides have equal lengths and all three internal angles are equal, each measuring 60 degrees. This type of triangle is not only symmetric but also holds distinct geometric properties that make calculations, such as determining the area, straightforward.

When dealing with such triangles, the formula for calculating the area is especially handy. Specifically, the area of an equilateral triangle with a side length of \( s \) is given by

• \[ A = \frac{\sqrt{3}}{4} s^2 \]

This formula arises from trigonometric considerations and symmetry, making it possible to express the area using only one variable: the side length.

Understanding how to compute the area of an equilateral triangle using this formula is essential for applying it in various mathematical problems and real-world scenarios involving triangular shapes.

Rate of Change

The concept of rate of change plays an important role in calculus and helps us understand how one quantity changes in relation to another. In this exercise, the rate of change we are concerned with is how the area of the equilateral triangle changes as its side length changes.

To derive this, we make use of differentiation, which allows us to find the rate at which a function changes at any point. With the formula for the area of an equilateral triangle,

• \[ A = \frac{\sqrt{3}}{4} s^2 \]

we identify that its rate of change with respect to the side length \( s \) is calculated by the derivative, which is

• \[ \frac{dA}{ds} = \frac{\sqrt{3}}{2} s \]

This derivative tells us how quickly the area \( A \) increases as the side length \( s \) increases.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. A derivative represents the rate at which a function changes and gives insights into its behavior.

In the context of this problem, we use differentiation to find the rate of change of the area of an equilateral triangle with respect to its side length. We start with the area formula,

• \[ A = \frac{\sqrt{3}}{4} s^2 \]

When we differentiate this formula with respect to \( s \), we obtain the derivative,

• \[ \frac{dA}{ds} = \frac{\sqrt{3}}{2} s \]

This derivative shows us that as the side length \( s \) changes, the area of the triangle also changes. In more applied terms, if we evaluate this derivative for a specific side length, such as \( s = 8 \) inches, we get

• \[ \frac{dA}{ds} = 4\sqrt{3} \]

This value represents how rapidly the area is increasing when each side of the triangle measures 8 inches.