Problem 51 - Prealgebra

Question

The width of a beach is changing at a rate of \(-9\) inches per year. How long will it take for the width of the beach to change \(-4.5\) feet?

Step-by-Step Solution

Answer

It will take 6 years for the beach's width to change by -4.5 feet.

Step 1: Understand the Problem

The problem asks us to find the time it will take for the width of a beach to change by a certain amount of feet, given that we know the rate of change in inches per year. We will need to convert the units so they are consistent.

Step 2: Convert Feet to Inches

We need to convert the change needed from feet to inches. Since 1 foot is 12 inches, we calculate \[ -4.5 \text{ feet} \times 12 \frac{\text{inches}}{\text{foot}} = -54 \text{ inches.} \]

Step 3: Use the Rate of Change

We know the rate of change is -9 inches per year. To find the time needed for a change of -54 inches, we can set up the equation:\[ \text{Time} = \frac{-54 \text{ inches}}{-9 \text{ inches per year}}. \]

Step 4: Solve the Equation

Plug the values into the equation \[ \text{Time} = \frac{-54}{-9} \] which simplifies to \[ \text{Time} = 6 \text{ years}. \]

Step 5: Interpret the Solution

The width of the beach will take 6 years to change by -4.5 feet, as the negative signs cancel each other out indicating a span of time.

Key Concepts

Rate of Change

The concept of "rate of change" is essential in understanding how quantities vary over time. It tells us how quickly or slowly a quantity is changing. For instance, in our exercise, the rate of change is given as

• -9 inches per year.

This negative sign indicates a decrease, meaning the beach is narrowing.
Understanding whether the rate is positive or negative helps us determine whether a quantity is increasing or decreasing. Rates of change can be found in various contexts, such as speed (miles per hour) or population growth (people per year).
In practice, if you are given a rate in different units from the final answer, unit conversion becomes necessary, as shown in our problem.

Problem Solving

Problem solving is the process of finding solutions to difficult or complex issues. It typically involves a series of organized steps, such as:

• Understanding the problem,
• Identifying what you know and what you need to find out,
• Breaking the problem into manageable parts,
• Executing a plan to reach a solution.

In the exercise, the problem was to determine how long it takes for the beach width to change by

• -4.5 feet,

while given a rate of change in inches per year.
By converting units and setting up equations, we break down the problem and make it easier to solve. Effective problem solving often means thinking critically and being methodical in approach.

Equations

Equations are fundamental tools in mathematics used to represent relationships between different quantities. They can be formed by setting an expression equal to a value or another expression. In our exercise, we used an equation to determine time:

• \[\text{Time} = \frac{-54 \text{ inches}}{-9 \text{ inches per year}}\]

This equation comes from the formula for rate:

• \[ \text{Rate} = \frac{\text{Change in Quantity}}{\text{Time}} \]

To find the time, we rearranged the formula to solve for time, which is the unknown in our case.
By substituting our known values into the equation, we solved it to find the solution. Equations provide a structured way to solve problems, making them crucial in problem-solving scenarios.