Question

Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour. (a) Build a model that expresses the distance \(d\) between the cars as a function of time \(t\). [Hint: At \(t=0,\) the cars are 2 miles south and 3 miles east of the intersection, respectively.] (b) Use a graphing utility to graph \(d=d(t) .\) For what value of \(t\) is \(d\) smallest?

Short answer

The distance function is \[ d(t) = \sqrt{ 2500t^2 - 360t + 13 } \]. Graph it to find the minimum distance value t.

Step-by-step solution

Define the positions of the cars as functions of time

Let the car moving from the south be Car A with its position given by \[ y(t) = 2 - 30t \] (since it starts 2 miles south and travels north at 30 miles per hour). For Car B moving from the east, its position is \[ x(t) = 3 - 40t \] (since it starts 3 miles east and travels west at 40 miles per hour).

Express the distance between the cars as a function of time

The distance between the cars at time t, using the Pythagorean Theorem, is given by \[ d(t) = \sqrt{x(t)^2 + y(t)^2} \] Substituting in the functions from Step 1, we get \[ d(t) = \sqrt{ (3 - 40t)^2 + (2 - 30t)^2 } \]

Simplify the distance function

Simplify the expression inside the square root: \[ d(t) = \sqrt{ (3 - 40t)^2 + (2 - 30t)^2 } = \sqrt{ 9 - 240t + 1600t^2 + 4 - 120t + 900t^2 } = \sqrt{ 2500t^2 - 360t + 13 } \]

Graph the distance function

Use a graphing utility or software to plot the function \[ d(t) = \sqrt{ 2500t^2 - 360t + 13 } \] Observe the graph to determine the minimum distance.

Find the minimum distance

By examining the graph, determine the value of t at which the distance d(t) is smallest. This can also be found using calculus techniques such as taking the derivative of d(t) and setting it to zero to find critical points.

Key concepts

distance function

In mathematical terms, a **distance function** determines the distance between two points at any given moment in time. In this exercise, the distance function measures how far apart two cars are as they travel towards an intersection. Since the cars are continuously moving, their positions change with time, which directly affects the distance between them.
To build the distance function, we first need to describe the cars' positions as functions of time. For Car A starting 2 miles south and heading north at 30 miles per hour, its position at time t is given by the function: \[ y(t) = 2 - 30t \]
For Car B starting 3 miles east and moving west at 40 mph, its position is: \[ x(t) = 3 - 40t \]
The distance between the two cars can be found using the formula that incorporates their positions: \[ d(t) = \textbackslash sqrt{(x(t)^2 + y(t)^2)}\] By plugging in the position functions of both cars, we get the distance function: \[ d(t) = \textbackslash sqrt{ (3 - 40t)^2 + (2 - 30t)^2 } \]

Pythagorean Theorem

The **Pythagorean Theorem** is a cornerstone in geometry, stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem is expressed as: \[\textbackslash a^2 + b^2 = c^2\]
In our exercise, the cars' positions at any time t span a right triangle, with one leg reflecting Car A’s position and the other leg reflecting Car B’s position. Here, the distance (d) is the hypotenuse of this hypothetical triangle, and can be encapsulated as: \[ d(t) = \textbackslash sqrt{(x(t)^2 + y(t)^2)}\]
This formula allows us to apply the Pythagorean Theorem to find the distance between the cars at any moment t, given their motion paths.

graphing utility

A **graphing utility** is an invaluable tool for visualizing mathematical functions and their behaviors. It allows you to see how a function changes over time or with different variable values. In our scenario, graphing the distance function \[ d(t) = \textbackslash sqrt{2500t^2 - 360t + 13}\] provides a visual representation of how the distance between the two cars varies with time.
To use a graphing utility effectively:

• Input the distance function accurately
• Observe where the function reaches its lowest point
• Analyze how different variables might affect the outcome

By graphing this function, you can easily identify the precise moment (value of t) when the cars are closest together.

minimizing distance

To **minimize distance** is to find the point at which the two objects (cars, in this case) are closest to each other. In practice, this involves pinpointing the minimum value of the distance function. There are two primary methods to achieve this:

• **Graphical Method:** As discussed, plotting the distance function on a graph will visually display where the shortest distance occurs. This is the minimum point on the curve.


• **Calculus Method:** Take the derivative of the distance function \[ f(t) = \textbackslash sqrt{2500t^2 - 360t + 13} \] and set it to zero to find critical points. Verify these points to locate the minimum distance.

Both methods effectively reveal the time (t) at which the cars are closest, simplifying the process of finding the minimum distance.