Problem 49 - College Algebra

Question

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the annual inflation rate, and \(S=\) the inflated value \(t\) years from now. Use this formula to solve, If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 65,000\) be worth in 10 years?

Step-by-Step Solution

Answer

The inflated value S of the house 10 years from now will be obtained by performing the calculations in the third step.

Step 1: Identify parameters given

In this exercise, the parameters given are the original value of the house, \( C = \$65,000 \), the annual inflation rate, \( r = 6\% = 0.06 \), and the time period in years, \( t = 10 \).

Step 2: Plug the parameters into the formula

By plugging values into the formula \( S=C(1+r)^{t} \), we get \( S= \$65,000 (1 + 0.06)^{10} \).

Step 3: Solve for the inflated value S

Calculate the inside of the parenthesese first due to order of operations, then raise it to the power of 10. After which, this result is multiplied by \$65,000. Perform these calculations to find the inflated value S.

Key Concepts

Annual Inflation Rate

The annual inflation rate is a crucial economic concept that represents the percentage increase in prices for goods and services over a year. This can erode your purchasing power.
When we say the inflation rate is 6%, it means that, on average, prices are increasing by 6% annually.
It’s important because it affects everything from the cost of living to your savings and investments.

• It is expressed as a percentage.
• It indicates how much more expensive a basket of goods and services becomes year over year.
• A higher annual inflation rate means prices are rising faster, which can impact economic planning.

By knowing the annual inflation rate, individuals and businesses can anticipate how much prices will rise and plan accordingly. In mathematical equations, like our inflation formula, it is often represented as a decimal. For example, a 6% inflation rate is used as 0.06 in formulas.

Exponential Growth

Exponential growth describes the process of increasing at a constant rate over a period of time. It is best visualized as a curve that becomes steeper over time, as opposed to linear growth, which proceeds by equal additions.
In the context of inflation, this means that the rate itself compounds, leading to larger increases as time progresses. In the formula \(S=C(1+r)^{t}\), we see exponential growth in action:

• \(C\) is your initial value, the cost of something today.
• \(r\) is the growth rate, or inflation rate.
• \(t\) is the time in years over which the value grows.
• \((1+r)^{t}\) shows how the original value grows exponentially each year.

Each year, the value doesn’t just increase by the same amount, but by an increasing amount because it’s growing on top of what’s already there, just like with interest. Understanding exponential growth is key to predicting future costs, whether you're looking at savings, investments, or inflation.

Order of Operations

Order of Operations is a fundamental concept in mathematics that determines the sequence in which mathematical operations are performed.
It ensures that calculations yield accurate results every time. In our equation \(S=C(1+r)^{t}\), adhering to the order of operations is essential for correct computation.
Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction):

• Parentheses: Solve operations inside parentheses first.
• Exponents: Solve powers and roots next.
• Multiplication/Division: Solve these operations from left to right.
• Addition/Subtraction: These final operations are also performed left to right.

In our scenario, this means you first calculate \((1+0.06)\). Then raise this result to the power of 10, and finally multiply by \(65,000\). Recognizing and applying the order of operations is crucial for solving any math problem accurately.