Question

An equilateral triangle initially has sides of length \(20 \mathrm{ft}\) when each vertex moves toward the midpoint of the opposite side at a rate of \(1.5 \mathrm{ft} / \mathrm{min}\). Assuming the triangle remains equilateral, what is the rate of change of the area of the triangle at the instant the triangle disappears?

Short answer

Answer: The rate of change of the area of the equilateral triangle at the moment it disappears is -30 * sqrt(3) ft^2/min.

Step-by-step solution

Find Side Length to Area Relationship

To determine the relationship between side length (s) and the area (A) of the equilateral triangle, we can use the formula: A = (s^2 * sqrt(3)) / 4 But we have to find the rate of change of A with time, so we need to have an equation with the time variable (t). We have the rate of change of side length (-1.5 ft/min), which means all three sides will decrease by 1.5 ft/min, so s(t) = 20 - 1.5t. Now, let's find the area in terms of time (t): A(t) = ((20 - 1.5t)^2 * sqrt(3)) / 4

Differentiate the Area Equation with respect to Time

Next, we'll differentiate A(t) with respect to time (t) to find the rate of change of the area: dA/dt = d((20 - 1.5t)^2 * sqrt(3) / 4) / dt Using the chain rule to differentiate this expression: dA/dt = (2*(20 - 1.5t) * (-1.5) * sqrt(3)) / 4 Simplify: dA/dt = -1.5(20 - 1.5t) * sqrt(3)

Find the Rate of Change of the Area when Triangle Disappears

The equilateral triangle disappears when its side length becomes 0. Thus, we need to find the time (t) when s(t) = 0: 20 - 1.5t = 0 t = 20 / 1.5 t = 40/3 minutes Now, we can substitute this time into our expression for dA/dt: dA/dt = -1.5(20 - 1.5(40/3)) * sqrt(3) dA/dt = -30 * sqrt(3) ft^2/min So, at the instant when the equilateral triangle disappears, the rate of change of its area is -30 * sqrt(3) ft^2/min.

Key concepts

Equilateral Triangle Area

Understanding the area of an equilateral triangle is an essential concept in geometry that also plays a crucial role in various calculus problems. An equilateral triangle is a three-sided polygon with all sides of equal length and all three interior angles equal to 60 degrees. The formula to compute the area (A) of such a triangle with side length (s) is \[ A = \frac{{s^2 \sqrt{3}}}{4} \].

This formula results from the fact that the height of an equilateral triangle forms a right-angled triangle with half of the side as its base and uses the Pythagorean theorem to solve for the height before calculating the area. For the exercise at hand, the starting side length of the equilateral triangle is given as 20 ft. When the vertices move inward, the side lengths change, and the area can be expressed as a function of time, leading to the discussion of related rates.

Related Rates

Related rates in calculus refer to finding the rate at which one quantity changes with respect to another, given a relationship between the two quantities. When two or more quantities are related by an equation, and one quantity is changing at a known rate, we can find the rate of change of another quantity using differentiation with respect to time.

In our exercise, as the vertices of the triangle approach the midpoint, the sides shrink at a known rate, precisely at -1.5 ft/min (negative since the sides are decreasing). This scenario exemplifies a related rates problem where we need to connect the rate of change of the side length with the rate of change of the area over time.

Differentiation

Differentiation is a fundamental operation in calculus used to find the rate at which a function is changing at any point. It gives us the derivative, which represents an instantaneous rate of change or the slope of the tangent line at any given point on a curve. When applying differentiation to a function of time, such as the area of a geometric shape, we obtain its rate of change with respect to time.

For instance, if the area (A) of a triangle is expressed in terms of time (t), differentiating A with respect to t will yield the rate of change of the area, denoted as \(\frac{dA}{dt}\). In the presented problem, by differentiating the time-dependent area function of the equilateral triangle, we determine the rate of change of that area as the side length changes over time.

Chain Rule

The chain rule is a differentiation technique that allows us to compute the derivative of composite functions. Essentially, when we have a function composed of multiple layers, the chain rule instructs us to differentiate the outer function and then multiply it by the derivative of the inner function. This rule becomes extremely useful in related rates problems, where an inside function, often a time-dependent variable, is nested within an outside function.

In our exercise, the square of the side length function \( (20 - 1.5t)^2 \) is nested within the area function. Applying the chain rule allows us to differentiate the area with respect to time accurately. The formula illustrates the derivative of the outer function, multiplies it by the derivative of the inner function \( -1.5 \) which is the rate of change of the side length, and incorporates the given \( \sqrt{3} \) factor arising from the original area formula.