Problem 62 - Calculus with Concepts in Calculus

Question

Two cars depart from the same location at the same time. One travels north at 40 miles per hour and the other travels east at 50 miles per hour. Find a formula for the function \(D\) that expresses in terms of \(t\) the distance between the cars \(t\) hours after departure.

Step-by-Step Solution

Answer

The distance function is \( D(t) = 10\sqrt{41} \cdot t \).

Step 1: Define the positions of the cars

The northbound car travels at a speed of 40 miles per hour. Thus, after \( t \) hours, its position is 40t miles north of the starting point. The eastbound car travels at a speed of 50 miles per hour, so after \( t \) hours, it is 50t miles east of the starting point.

Step 2: Apply the Pythagorean theorem

To find the distance between the two cars, we can treat the positions of the cars as points on a coordinate plane, creating a right triangle where the legs are the distances traveled north and east. The northbound car's distance of 40t miles and the eastbound car's distance of 50t miles are the legs of this triangle.

Step 3: Write the distance formula using Pythagorean theorem

According to the Pythagorean theorem, the distance \( D \) between the two cars is the hypotenuse of a right triangle, calculated as \( D = \sqrt{(40t)^2 + (50t)^2} \).

Step 4: Simplify the distance formula

Simplifying the expression for \( D \), we have: \( D = \sqrt{1600t^2 + 2500t^2} = \sqrt{4100t^2} \). Simplifying further, \( D = \sqrt{4100} \cdot t \). The numerical factor can be written as \( \sqrt{4100} = 10\sqrt{41} \). Thus, the distance is \( D = 10\sqrt{41} \cdot t \).

Step 5: Finalize the function

Finally, we express the distance \( D \) as a function of time \( t \). The formula is: \( D(t) = 10\sqrt{41} \cdot t \).

Key Concepts

Distance Function

The distance function is a way to calculate the separation between two points. In this scenario, we are finding the distance between two cars moving in perpendicular directions: one traveling north and the other east. The distance function helps us understand how far apart they are at any given time, denoted as \( t \), after they start from the same location.
To put it simply, the distance function gives the hypotenuse of a right triangle formed by these two paths. This formula is expressed as \( D(t) = 10\sqrt{41} \cdot t \). It shows us how the distance changes over time. Whenever you come across problems like this, look for distances covered and travel directions to apply the distance function effectively.

Coordinate Plane

A coordinate plane is a two-dimensional surface where you can represent geometric figures with coordinates. In our case, we use the coordinate plane to plot the starting point of both cars, their pathways, and how they create a right angle with each other.
Imagine placing a grid over a map, where:

• The northbound car’s position is on the vertical line (y-axis).
• The eastbound car’s position is on the horizontal line (x-axis).

By visualizing this on a coordinate plane, you can easily see the straight line distance between these two points. This method is useful to determine clear distances and directions, simplifying the calculation process.

Right Triangle

A right triangle is a key concept when working with the Pythagorean theorem. It's formed when two shorter sides (the legs) and one longest side (the hypotenuse) create a 90-degree angle between them.
In this problem, the right triangle forms between the distances traveled by each car. The northbound car’s 40t miles and the eastbound car’s 50t miles act as the legs of the triangle.
The hypotenuse is the distance we want to find.

• The Pythagorean theorem tells us that the sum of the squares of the legs is equal to the square of the hypotenuse.
• It provides a reliable method to calculate distances, reinforcing understanding of geometric relationships.

Understanding right triangles is crucial when solving real-world problems involving angles and distances.

Mathematical Formula

Mathematical formulas provide a precise way to depict relationships between quantities. In this exercise, we used the Pythagorean theorem to derive a formula. This formula, \( D(t) = 10\sqrt{41} \cdot t \), tells us the distance between two moving objects over time.
Let's break down how we got here:

• Calculate each car’s traveled distance as function of time (40t and 50t).
• Apply the Pythagorean theorem: \( D = \sqrt{(40t)^2 + (50t)^2} \).
• Simplify to get \( D(t) = 10\sqrt{41} \cdot t \).

Formulas like these are the backbone of problem solving in math, offering a clear path from a scenario to a solvable equation. Mastering formula manipulation ensures effective understanding and application in diverse scenarios.