Problem 59 - Algebra 2

Question

UTFRAIURE In one of Grimm's Fairy Tales, Rumpelsititskin has the ability to spin straw into gold. Suppose on the first day, he spun 5 pieces of straw into gold, and each day thereafter he spun twice as much. How many pieces of straw would he have spun into gold by the end of the week? (lesson \(11-4\) )

Step-by-Step Solution

Answer

635 pieces.

Step 1: Understand the Problem

Rumpelstiltskin is spinning 5 pieces of straw into gold on the first day. Each subsequent day, he spins twice as many pieces as the previous day. The goal is to determine the total number of pieces spun into gold by the end of a week, which is 7 days.

Step 2: Calculate Daily Straw Count

We need to determine how many pieces of straw Rumpelstiltskin spins each day. Starting with 5 on Day 1, then multiplying each subsequent day by 2:- Day 1: 5 pieces- Day 2: 5 \( \times \) 2 = 10 pieces- Day 3: 10 \( \times \) 2 = 20 pieces- Day 4: 20 \( \times \) 2 = 40 pieces- Day 5: 40 \( \times \) 2 = 80 pieces- Day 6: 80 \( \times \) 2 = 160 pieces- Day 7: 160 \( \times \) 2 = 320 pieces

Step 3: Calculate Total Straw by Week's End

Sum the daily amounts to get the total number of pieces by the end of the week:\[5 + 10 + 20 + 40 + 80 + 160 + 320\]

Step 4: Solve the Total Sum

Compute the sum of all daily pieces over the week:\(5 + 10 + 20 + 40 + 80 + 160 + 320 = 635\).

Step 5: Answer the Question

By the end of the week, Rumpelstiltskin would have spun a total of 635 pieces of straw into gold.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept in mathematics where a quantity increases by a constant multiple over equal intervals of time. This means that instead of adding a fixed amount, we multiply by a fixed factor to find the next term.
In the story of Rumpelstiltskin, each day Rumpelstiltskin spins twice as many pieces of straw as the previous day. This doubling effect is a classic example of exponential growth. Some key characteristics of exponential growth include:

• Rapid increase over time: As we observe in this exercise, starting from a small number like 5, the daily totals quickly balloon to 320 pieces by the end of the week.
• Geometric sequence: The numbers 5, 10, 20, 40, and so on form a geometric sequence where each term is a multiple of the previous one.
• Doubling period: In this case, the 'doubling' of straw spun from one day to the next is direct and observable.

Understanding exponential growth helps explain many natural and theoretical processes in mathematics and real life, such as population growth, viral spread, and compound interest.

Problem-Solving in Mathematics

Problem-solving is a crucial part of mathematics and involves approaching problems methodically to find solutions. Let's consider how it looks in our Rumpelstiltskin exercise. Key steps in mathematical problem-solving include:

• Understanding the Problem: First, we identified that the number of straw pieces Rumpelstiltskin spins grows exponentially, doubling each day.
• Breaking Down the Problem: By calculating piece counts day by day, we can see how the growth builds and better grasp the overall pattern.
• Finding a Strategy: We saw that adding each daily result gives us the total number spun over the week.
• Checking and Re-Evaluating: It’s vital to verify our calculations, ensuring each step logically follows and exploring if there’s a quicker route to the solution.

These steps help students tackle a variety of mathematical challenges beyond this exercise, setting a strong foundation for critical thinking and logical reasoning.

Algebraic Expressions

Algebraic expressions play a vital role in organizing and solving problems that involve patterns like those seen in Rumpelstiltskin's straw spinning.In this exercise, we used simple math to calculate daily totals. However, for more complex scenarios, algebra steps in with expressions to define patterns more succinctly. Algebraic expressions for geometric sequences are very handy.For the Rumpelstiltskin sequence:

• The number of straw spun each day forms the sequence: 5, 10, 20, ... This can be expressed as: 5, 5\( \times \)2, 5\( \times \)2^2, 5\( \times \)2^3, ..., up to the 7th day.
• The general formula here is based on exponential growth: 5\( \times \)2^{(n-1)}, where n is the day number.
• Having a general formula allows calculating the total or any specific day’s count without individually multiplying each step.

Using algebra, we can structure solutions more efficiently and gain insights into broader patterns and applications in mathematics, making it a powerful tool for mathematical reasoning.