Problem 5 - Algebra 2

Question

WEATHER Heavy rain caused a river to rise. The river rose three inches the first day, and each day it rose twice as much as the previous day. How much did the river rise in five days?

Step-by-Step Solution

Answer

The river rose 93 inches in five days.

Step 1: Understand the Pattern

The river's rise follows a geometric sequence where each day's rise is twice the rise of the previous day. Starting from the first day, the rise on the first day is 3 inches, the second day is twice that (6 inches), and this pattern continues for each subsequent day.

Step 2: Calculate Each Day's Rise

Let's compute the rise in the river for each of the five days using the sequence: * Day 1: 3 inches * Day 2: 2 times 3 = 6 inches * Day 3: 2 times 6 = 12 inches * Day 4: 2 times 12 = 24 inches * Day 5: 2 times 24 = 48 inches

Step 3: Sum Up the Rises

Add up the river's rise for each day to get the total rise over the five days: \(3 + 6 + 12 + 24 + 48 = 93\) inches.

Key Concepts

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the exercise, you see such a pattern with the rise of the river. The river's rise each day forms a sequence: 3 inches, 6 inches, 12 inches, 24 inches, and 48 inches. The common ratio here is 2, as each day's rise is twice that of the previous day.

It's important to understand this concept, as it forms the basis of many mathematical models. Some properties of geometric progressions include:

• The ratio between consecutive terms is constant.
• They can either grow larger (as in our river example) or smaller, depending on whether the common ratio is greater or less than one.
• The general form of a geometric sequence is given by the formula: \( a_n = a_1 \, r^{n-1} \), where \(a_1\) is the first term and \(r\) is the common ratio.

Recognizing such sequences in real-world problems can help you simplify and solve complex challenges effectively.

Exponential Growth

Exponential growth describes the increase of a quantity where the rate of growth is proportional to the current size of the quantity. This concept is vividly illustrated in the example with the river rise.

Each day's increase is doubling the amount from the previous day, a hallmark of exponential growth.

• The key feature is the rapid increase over time. In only five days, the rise went from 3 inches to a total of 93 inches.
• Exponential growth is often modeled with the formula: \( y = a \cdot e^{kt} \), where \( a \) is the initial amount, \( e \) is the base of natural logarithms, \( k \) is the growth constant, and \( t \) is time.
• In simpler geometric growth, as seen in this scenario, the formula simplifies to using a constant multiplier, like our common ratio of 2.

Understanding this concept is crucial, as exponential patterns appear frequently in fields such as finance, biology, and physics.

Algebra Concepts

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It helps to present relationships and solve problems efficiently. In the exercise, algebraic thinking helps us:

Let's consider how algebra is used in handling geometric sequences. The task required determining the pattern of the river rise and then calculating the sum of these terms.

• By expressing the daily rise as a sequence using algebra, you simplified the problem of adding up multiple terms.
• The ability to recognize a geometric progression, express it algebraically, and sum it up is a key skill in algebra.
• Another important algebraic skill here is the simplification: adding multiple terms into a single total rise of 93 inches over five days.

Practicing these concepts makes it easier to tackle similar problems, translating real-world situations into solvable equations easily.