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There are many stories in children's literature that involve magic pots. An Italian variation goes something like this: A woman puts a pot of water on the stove to boil. She says some special words, and the pot begins filling with pasta. Then she says another set of special words, and the pot stops filling up. Suppose someone overhears the first words, takes the pot, and starts it in its pasta-creating mode. Two liters of pasta are created. Then the pot continues to create more pasta because the impostor doesn't know the second set of words. The volume continues to increase \(50 \%\) per minute. a. Write an equation that models the amount of pasta in liters, \(y\), after \(x\) minutes. (a) b. How much pasta will there be after 30 seconds? c. How much pasta will there be after 10 minutes? d. How long, to the nearest second, will it be until the entire house, which can hold 450,000 liters, is full of pasta?

Step by step solution

01

Understand the Problem

The problem describes a situation where a pot is continuously creating pasta at a rate that increases by 50% every minute after starting with an initial 2 liters of pasta.

02

Create an Exponential Growth Model

Given that the pasta increases by 50% per minute, we can write the equation for this compound growth as:\[ y = 2 \times (1.5)^x \]where \( y \) is the amount of pasta in liters after \( x \) minutes.

03

Calculate Pasta after 30 Seconds

Since 30 seconds is half a minute, we must consider the growth for half a minute. Using the formula:\[ y = 2 \times (1.5)^{0.5} \approx 2 \times 1.2247 = 2.4494 \text{ liters} \]

04

Calculate Pasta after 10 Minutes

Substitute \( x = 10 \) into the growth equation:\[ y = 2 \times (1.5)^{10} \approx 2 \times 57.665 = 115.33 \text{ liters} \]

05

Determine Time to Fill the House

We need to find \( x \) such that \( y = 450,000 \). Using the growth equation:\[ 450,000 = 2 \times (1.5)^x \]Divide both sides by 2:\[ 225,000 = (1.5)^x \]Take the logarithm of both sides and solve for \( x \):\[ x = \frac{\log(225,000)}{\log(1.5)} \approx 25.09 \]Convert to seconds: \( 25.09 \times 60 \approx 1505 \text{ seconds} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Modeling

When faced with a real-world problem, we often use mathematical modeling to create a simplified representation of the natural scenario. This allows us to predict and analyze behaviors using mathematical equations and functions. In the case of the magic pasta pot, the goal is to model how pasta fills up in the pot over time. We begin with 2 liters, and the amount increases exponentially by 50% every minute.

The aid of mathematical models like exponential functions in such scenarios lets us transform a narrative problem into a set of solvable mathematical statements. By identifying that the pasta increases at a consistent rate (50% per minute), we use this information to frame the problem as an exponential growth situation. From there, we can calculate future pasta amounts at various times or determine how long it will take to reach a specific pasta volume.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In our magic pot scenario, algebra helps to translate the situation into a mathematical equation that models exponential growth. The core equation given is:\[ y = 2 \times (1.5)^x \]

Here, \(y\) represents the total amount of pasta made in liters after \(x\) minutes. The starting amount of pasta is 2 liters (which appears as a constant in the equation), and it grows by a factor of 1.5 every minute. Each step of algebraic manipulation, such as solving the equation to find \(y\) or working out the value of \(x\) when \(y\) is a known target (like filling the house), is carried out by applying algebraic operations.

• You use algebra to substitute values into the equation for calculations.
• Algebra allows you to rearrange and solve equations to find unknown values, such as the time in minutes to fill the house.

Exponential Functions

Exponential functions are mathematical models used to describe growth or decay processes that happen at a constant percentage rate per unit time. In our problem, the pasta amount grows exponentially because it increases by 50% every minute.

The general form of an exponential growth function is:\[ y = a \times b^x \]

In this form, \(a\) is the initial amount, \(b\) is the growth factor (1 plus the growth rate), and \(x\) is time. For the magic pot exercise:

• \(a = 2\): This is our initial condition, starting with 2 liters.
• \(b = 1.5\): This represents the 50% growth rate since a 50% increase means multiplying by 1.5.
• The time \(x\) is measured in minutes.

Understanding and applying exponential functions lets us predict future amounts of pasta efficiently and determine how fast it will fill spaces of known sizes, like the entire house.