Question

A Ford Focus and a Freightliner Cascadia truck leave an intersection at the same time. The Focus heads east at an average speed of 60 miles per hour, while the Cascadia heads south at an average speed of 45 miles per hour. Find an expression for their distance apart \(d\) (in miles) at the end of \(t\) hours.

Short answer

The distance apart is 75t miles.

Step-by-step solution

Understand the direction and speed

The Ford Focus travels east at 60 mph and the Freightliner Cascadia travels south at 45 mph. To find the distance apart after a certain time, visualize the problem as a right triangle, where the paths of the vehicles form the legs of the triangle.

Define the positions after time t

After t hours, the Ford Focus will have traveled a distance of 60t miles east, and the Freightliner Cascadia will have traveled a distance of 45t miles south.

Use the Pythagorean theorem

The distance apart d at the end of t hours is the hypotenuse of a right triangle with legs of 60t and 45t. According to the Pythagorean theorem, the distance d can be expressed as: \[d = \sqrt{(60t)^2 + (45t)^2}\]

Simplify the expression

Simplify the expression to find the distance d:\[d = \sqrt{3600t^2 + 2025t^2} = \sqrt{5625t^2} = 75t\]

Key concepts

right triangle

In this exercise, understanding the concept of a right triangle is crucial. A right triangle has one angle exactly equal to 90 degrees. The sides of this triangle are called the legs and the hypotenuse. The hypotenuse is the side opposite the right angle and is always the longest.

When the Ford Focus and the Freightliner Cascadia travel in perpendicular directions—east and south respectively—they form the legs of a right triangle. Their paths create a big right triangle, with each vehicle's travel representing one leg. This helps us visualize the problem and apply mathematical principles like the Pythagorean theorem to find the solution.

vehicle motion

To solve the problem, you need to understand the motion of vehicles. Here, the Ford Focus travels east at 60 miles per hour (mph), and the Freightliner Cascadia heads south at 45 mph.

Both vehicles start from the same intersection simultaneously. Over time, each vehicle covers more distance, and the distance between them increases. The Ford Focus moves in a straight line toward the east, while the Freightliner Cascadia moves straight toward the south. This constant motion at fixed speeds makes calculating their separation distance simpler.

distance formula

The distance formula is vital to understanding this problem. The formula for calculating the distance between two points in a coordinate plane is: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This formula derives from the Pythagorean theorem.

In our exercise, after t hours, the Ford Focus has traveled 60t miles east, and the Freightliner Cascadia has traveled 45t miles south. These distances become the coordinates on our right triangle. Plugging these values into the distance formula gives us: \[ d = \sqrt{(60t)^2 + (45t)^2} \]. Simplified, it becomes: \[ d = \sqrt{3600t^2 + 2025t^2} = \sqrt{5625t^2} = 75t \].

speed and time relationship

Understanding the relationship between speed, distance, and time is essential. It is summarized by the formula: Distance = Speed × Time.

For the Ford Focus, traveling east at 60 mph: Distance_Focus = 60 × t miles. For the Freightliner Cascadia, traveling south at 45 mph: Distance_Cascadia = 45 × t miles.

Over any given time (t hours), the distances these vehicles travel can be expressed in terms of their respective speeds and time. This time-speed relationship is a fundamental concept for determining how far each vehicle moves and, subsequently, their distance apart over time.