Problem 15 - Algebra 2 and Trigonometry

Question

When Rita was five, she had \(\$ 1\) in her piggy bank. The next year she doubled the amount that she had in her piggy bank to \(\$ 2 .\) She decided that each year she would double the amount in her piggy bank. How old will Rita be when she has at least \(\$ 1,000\) in her piggy bank?

Step-by-Step Solution

Answer

Rita will be 15 years old when she has at least $1000 in her piggy bank.

Step 1 - Understanding the Problem

Rita starts with $1 in her piggy bank at age five. Each year, she doubles the amount in the piggy bank. We need to find out at what age the amount in her piggy bank will be at least $1000.

Step 2 - Setting Up the Equation

The general formula for the amount of money in the piggy bank after each year is given by: \[ A(n) = 1 \times 2^{n-5} \]where \(n\) is Rita's age. We want to find \(n\) such that:\[ 2^{n-5} \geq 1000 \]

Step 3 - Solving the Inequality

Let's solve for \(n\) in the inequality \(2^{n-5} \geq 1000\). Taking the logarithm base 2 of both sides gives:\[ n-5 \geq \log_{2}{1000} \]Approximating \(\log_{2}{1000} \), we get:\[ n-5 \geq 9.965 \]

Step 4 - Calculating Rita's Age

We solve for \(n\) in the inequality:\[ n \geq 14.965 \]Since \(n\) must be a whole number (as age is an integer), we round up to get:\[ n = 15 \]

Step 5 - Concluding

When \(n=15\), it means Rita is 15 years old. Therefore, she will have at least $1000 in her piggy bank when she is 15.

Key Concepts

Exponential Function

Exponential functions are a fascinating and powerful tool in mathematics. They describe situations where something grows at a consistent rate over time. In our example with Rita's piggy bank, the amount of money doubles each year, which is a perfect illustration of exponential growth. In general, an exponential function is written as \( f(x) = a \cdot b^x \), where \( a \) is the initial amount, \( b \) is the base that represents the growth factor, and \( x \) is the variable representing time or another independent variable.

• If \( b > 1 \), the quantity grows over time. This is known as exponential growth.
• If \( b < 1 \), the quantity decreases over time, which is called exponential decay.

In the context of Rita's savings, each year the money in her piggy bank is multiplied by 2, so the base \( b = 2 \). Hence, the function representing the amount in her piggy bank becomes \( A(n) = 1 \cdot 2^{n-5} \), with \( n \) being her age. Understanding how exponential functions work helps us predict future values accurately.

Logarithms

Logarithms are the inverse operation of exponentiation, much like subtraction is the inverse of addition. If you have an exponential equation like \( b^y = x \), then the logarithm helps us solve for \( y \). Specifically, \( y = \log_b{x} \) where \( b \) is the base and \( x \) is the argument of the logarithm.

• Logarithms are useful for solving equations where the unknown variable is an exponent.
• In this exercise, we needed to find how many years Rita would take to reach a certain amount in her piggy bank. This was done using a logarithm.

When Rita wants at least $1000, we set up the inequality \( 2^{n-5} \geq 1000 \). To solve this for \( n \), we take the logarithm base 2 of both sides, resulting in \( n-5 \geq \log_2{1000} \). This gives us a practical way to approach the problem beyond simple guesswork.

Inequality Solving

Solving inequalities is a crucial skill in mathematics. Unlike equations that require one true value, inequalities allow for a range of values. When solving \( 2^{n-5} \geq 1000 \), we are finding all possible ages \( n \) where Rita's savings will be at least $1000.

• Inequalities can be solved similarly to equations, but we have to remember that multiplying or dividing by a negative number reverses the inequality sign.
• In this situation, after taking the logarithm of both sides, we end up with \( n-5 \geq 9.965 \), which we then solve by isolating \( n \).

For Rita's problem, because \( n \) refers to an age, it must be a whole number. Therefore, we round up from 14.965 to 15 to get a realistic answer. Being able to interpret and solve inequalities allows us to deal with scenarios where results are not limited to a singular finite answer.