Problem 31 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Two sisters, Nina and Lori, part on a street corner. Lori saunters due north at a rate of 150 feet per minute and Nina jogs off due east at a rate of 320 feet per minute. Assuming they maintain their speeds and directions, express the distance between the sisters as a function of the number of minutes since they parted.

Step-by-Step Solution

Answer

The distance between Lori and Nina as a function of the time since they parted is given by \(D(t) = t \sqrt{150^2 + 320^2}\) feet.

Step 1: Identify Variables

Let's denote the distance traveled by Lori as \(L(t)\) and that traveled by Nina as \(N(t)\). Here, \(t\) is the number of minutes since they parted. Lori moves at a rate of 150 feet per minute and Nina moves at a rate of 320 feet per minute.

Step 2: Formulate Distance Function for Each Sister

Based on their speeds, the distance function would be: \(L(t) = 150t\) for Lori and \(N(t) = 320t\) for Nina. These equations mean that each minute, Lori and Nina add 150 and 320 feet to their total distance respectively.

Step 3: Apply Pythagorean Theorem

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle. Let \(D(t)\) denote the distance between Lori and Nina at time \(t\). According to the Pythagorean theorem, we have \(D(t)^2 = L(t)^2 + N(t)^2\). Substituting the expressions for \(L(t)\) and \(N(t)\), we get \(D(t)^2 = (150t)^2 + (320t)^2\).

Step 4: Find the Distance Function

To find the function for distance \(D(t)\) we take the square root of both sides of our equation. Thus, the distance function between the sisters becomes: \(D(t) = \sqrt{(150t)^2 + (320t)^2}\). Simplifying this we get \(D(t) = t \sqrt{150^2 + 320^2}\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that helps us find the length of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the 90-degree angle) is equal to the sum of the squares of the other two sides. This relationship is expressed as: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.
In the given exercise, Lori and Nina's paths form the two shorter sides of a right triangle, with the distance between them being the hypotenuse. By substituting their distances into the formula, we can find how far apart they are over time.

Rate of Change

The rate of change is a concept used to describe how one quantity changes in relation to another. In this context, it refers to how fast the distance each sister travels changes per unit of time.
When Lori walks north at 150 feet per minute and Nina jogs east at 320 feet per minute, these speeds represent their rates of change. Knowing these rates is essential for determining how far they travel over time.

• Lori's rate: 150 feet/minute
• Nina's rate: 320 feet/minute

By understanding their rates, we can form the distance functions for each sister, which are crucial for calculating the total distance between them.

Distance Formula

The distance formula helps us to compute the distance between two points in a coordinate plane. However, in our exercise, we are calculating a dynamic scenario, which means the distance between Lori and Nina changes as they keep moving.
To find this distance over time, we express it as a function, \( D(t) \), which involves not only their speeds but also the Pythagorean Theorem. The formula for \( D(t) \) becomes: \[ D(t) = \sqrt{(150t)^2 + (320t)^2} \]

Breaking it Down:

- Calculate the distance Lori travels north as \( 150t \). - Calculate the distance Nina travels east as \( 320t \). - Use the Pythagorean theorem to account for both travels as right triangle sides and find the hypotenuse, which represents the distance between them.

Functions

A function is a relationship between two variables, typically describing how one variable depends on the other. In simple terms, it shows how different inputs give different outcomes.
In our exercise, we create functions to describe how the distance each sister travels varies over time. The functions we'd use are:

• \( L(t) = 150t \): for Lori, indicating her distance in feet over time \( t \).
• \( N(t) = 320t \): for Nina, indicating her distance in feet over time \( t \).
• \( D(t) \): the function representing the hypotenuse or the direct distance between them.

These functions simplify complex movements into manageable equations, helping us comprehend and compute their evolving distances effectively.