Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve by writing a sum of signed numbers and adding. The water level of a reservoir is measured over a five-month period. At the beginning, the level is 20 feet. During this time, the level rose 3 feet, then fell 2 feet, then fell 1 foot, then fell 4 feet, and then rose 2 feet. What is the reservoir's water level at the end of the five months?

Step-by-Step Solution

Answer

The reservoir's water level at the end of the five months is 18 feet.

Step 1: Identify the Initial Level

The initial water level of the reservoir is 20 feet.

Step 2: Account for Changes

Identify all the increases and decreases in the reservoir's water level over the five-month period. - Increase by 3 feet - Decrease by 2 feet - Decrease by 1 foot - Decrease by 4 feet - Increase by 2 feet

Step 3: Write Down the Sum of Signed Numbers

Write a sum of signed numbers to reflect the changes in the water level. A positive number indicates a rise in the water level, while a negative number indicates a fall. This can be written as: \( 20 + 3 - 2 - 1 - 4 + 2 \)

Step 4: Perform the Calculation

Perform the addition and subtraction calculations to find the water level at the end of the five months. This gives: \( 20 + 3 - 2 - 1 - 4 + 2 = 18 \) feet

Key Concepts

Signed Numbers

Understanding signed numbers is crucial since they help us operate with both positive and negative values in math. A signed number contains a "+" or a "-" symbol. Positive numbers increase the value, while negative numbers decrease it. When dealing with real-world problems, signed numbers can represent things like elevation changes or financial gains and losses.
For example:

• "+3" means a rise or increase of 3 units.
• "-2" means a drop or decrease of 2 units.

In the context of the problem about the reservoir, each change in the water level is represented as a signed number:

Addition and Subtraction

Addition and subtraction are fundamental operations in math that allow us to calculate total values and differences. Combining signed numbers allows us to track cumulative changes over time.
When adding numbers together, treat them as cumulative increases or decreases.
For instance:

• If we add a positive number, it means we are increasing the total value.
• Adding a negative number is akin to subtracting its absolute value, which reduces the total.

Consider the sum in the reservoir problem:

• Start with 20 (initial level).
• Add 3 (indicating a rise), so it becomes 23.
• Subtract 2 (fall), it becomes 21.
• Subtract another 1, it becomes 20.
• Subtract 4 more, resulting in 16.
• Finally, add 2, ending with 18 feet.

Problem Solving

Effective problem solving in math requires identifying what is given and what needs to be found. Start by understanding the problem: we're tracking changes over five months to find the final water level.
Here's a simple approach:

• Identify the initial value. For the reservoir, it's 20 feet.
• Break down each subsequent change into a signed number (+3, -2, -1, -4, +2).
• Create a sum of these signed numbers combined with the initial value.
• Calculate step by step, ensuring each operation is clear.

Following these principles can help solve various problems that involve tracking changes or cumulative quantities. By calculating with accuracy, you'll find solutions easily.