Question

An oil slick from a runaway offshore oil well is advancing toward a beach 354 miles away at the rate of 10.5 mi/day. Two days after the spill, cleanup ships leave the beach and steam toward the slick at a rate of 525 milday. At what distance from the beach will they reach the slick?

Short answer

The cleanup ships will meet the oil slick at a distance of 315 miles from the beach.

Step-by-step solution

Determine the distance covered by the oil slick before ships depart

Calculate the distance the oil slick has moved in two days using the rate of the oil spill's movement. The formula to calculate the distance is Distance = Speed × Time. Here, the speed is 10.5 mi/day and the time is 2 days.

Calculate the relative speed of the ships towards the oil slick

Since the cleanup ships are moving toward the slick and both are moving in opposite directions, their speeds will be added to obtain the relative speed. The relative speed is the sum of the speed of the cleanup ships and the speed of the oil slick.

Determine the remaining distance for the ships to meet the slick

Subtract the distance already covered by the oil slick from the total distance to the beach to find the remaining distance that the ships have to cover to meet the slick.

Calculate the time it will take for the ships to reach the oil slick

Use the formula Time = Distance / Speed with the relative speed obtained in Step 2 and the remaining distance from Step 3 to calculate the time it will take for the ships to reach the slick.

Calculate the distance from the beach where the ships will meet the slick

Multiply the speed of the cleanup ships by the time calculated in Step 4 to get the distance from the beach where the ships will meet the slick.

Key concepts

Understanding the Distance Formula

The distance formula is a simple, yet powerful tool in physics and mathematics, used to calculate how far an object has traveled over a certain period of time. It is essentially an application of the concept of speed as a rate of change. The standard formula is expressed as \( Distance = Speed \times Time \).

In the given exercise, we apply this formula to determine how far the oil slick moved in two days. Our rate of change is the slick's advancing speed at 10.5 mi/day, and time is 2 days, leading us to a computation of \( 10.5 \text{ mi/day} \times 2 \text{ days} = 21 \text{ miles} \). This calculation gives us a crucial piece of data for later steps, as it represents the progress the slick has made before any cleanup effort has begun.

Rate of Change in Relative Motion

When two objects are moving in relation to one another, we often measure their motion using the concept of relative speed. Relative speed becomes particularly important when the objects move in opposing directions, as is the case with the oil slick and the cleanup ships in the exercise.

In this context, the relative speed is found by adding the speed of both entities moving towards one another. With the cleanup ships steaming at 525 mi/day and the oil slick moving at 10.5 mi/day, the relative speed is calculated as \( 525 \text{ mi/day} + 10.5 \text{ mi/day} = 535.5 \text{ mi/day} \). This is the effective speed at which the distance between the ships and the slick is closing. Understanding this concept is crucial for predicting encounters or collisions between objects in motion.

Calculating Time: The Key to Precise Prediction

Time calculations are at the heart of motion problems, helping predict when and where events will occur. In our original exercise with the oil slick, we need to determine how long it will take for the cleanup ships to reach the slick. For this, we use the formula \( Time = Distance / Speed \).

First, we ascertain the remaining distance to be covered by subtracting the distance already traveled by the slick from the total distance to the beach. This remaining distance, divided by the relative speed we calculated, yields the time required for the ships to meet the slick. A precise grasp of time calculations allows us to plan and execute timely interventions in real-life scenarios such as environmental clean-ups or coordinating logistics.