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Oct 13, 2025 by windailys.com 96 views

Tessa's Terrific Tees: A Mathematical Analysis of the T-Shirt Printing Machine's Depreciation

Tessa is excited to launch her own custom T-shirt business, Tessa's Terrific Tees. As part of her startup costs, she invested in a T-shirt printing machine valued at $2,200. However, the machine's value depreciates over time, and Tessa needs to understand the rate at which it loses its value. In this article, we will analyze the depreciation of the T-shirt printing machine using exponential functions.

The machine loses $\frac{1}{4}$ of its value every year. This means that the remaining value of the machine after one year is $1 - \frac{1}{4} = \frac{3}{4}$ of its original value. To find the value of the machine after $n$ years, we can use the formula:

V(n) = V_0 \left(1 - \frac{1}{4}\right)^n

where $V_0$ is the initial value of the machine ($2,200) and $n$ is the number of years.

The formula above is an example of an exponential function, which has the general form:

f(x) = ab^x

where $a$ and $b$ are constants, and $x$ is the variable. In our case, $a = V_0 = 2,200$ and $b = 1 - \frac{1}{4} = \frac{3}{4}$ .

We can simplify the formula by substituting the values of $a$ and $b$ :

V(n) = 2,200 \left(\frac{3}{4}\right)^n

This formula gives us the value of the machine after $n$ years.

To find the value of the machine after $n$ years, we can plug in the value of $n$ into the formula:

V(n) = 2,200 \left(\frac{3}{4}\right)^n

For example, if we want to find the value of the machine after 5 years, we can plug in $n = 5$ :

V(5) = 2,200 \left(\frac{3}{4}\right)^5

Using a calculator, we can evaluate this expression:

V(5) \approx 1,071.43

This means that after 5 years, the value of the machine will be approximately $1,071.43.

To visualize the depreciation of the machine, we can graph the function:

V(n) = 2,200 \left(\frac{3}{4}\right)^n

Using a graphing calculator or software, we can plot the graph of this function. The graph will show the value of the machine over time, with the value decreasing exponentially.

In this article, we analyzed the depreciation of Tessa's T-shirt printing machine using exponential functions. We derived a formula for the value of the machine after $n$ years and used it to calculate the value after 5 years. We also graphed the depreciation of the machine to visualize the rate at which it loses its value. This analysis will help Tessa understand the financial implications of owning a T-shirt printing machine and make informed decisions about her business.

There are several ways to extend this analysis. For example, we could:

Investigate the impact of inflation: Inflation can affect the value of the machine over time. We could investigate how inflation affects the depreciation of the machine.
Compare different depreciation rates: We could compare the depreciation rate of the machine to other machines or assets to see how it compares.
Develop a model for the machine's lifespan: We could develop a model for the machine's lifespan, taking into account factors such as usage and maintenance.

These are just a few ideas for future work. The analysis of the depreciation of Tessa's T-shirt printing machine is a rich area of study, and there are many ways to extend and refine this analysis.
Tessa's Terrific Tees: A Q&A on the T-Shirt Printing Machine's Depreciation

In our previous article, we analyzed the depreciation of Tessa's T-shirt printing machine using exponential functions. We derived a formula for the value of the machine after $n$ years and used it to calculate the value after 5 years. We also graphed the depreciation of the machine to visualize the rate at which it loses its value. In this article, we will answer some frequently asked questions about the depreciation of the machine.

A: The machine loses $\frac{1}{4}$ of its value every year. This means that the remaining value of the machine after one year is $1 - \frac{1}{4} = \frac{3}{4}$ of its original value.

A: To calculate the value of the machine after $n$ years, you can use the formula:

V(n) = 2,200 \left(\frac{3}{4}\right)^n

where $V(n)$ is the value of the machine after $n$ years, and $n$ is the number of years.

A: Using the formula above, we can calculate the value of the machine after 5 years:

V(5) = 2,200 \left(\frac{3}{4}\right)^5

Using a calculator, we can evaluate this expression:

V(5) \approx 1,071.43

This means that after 5 years, the value of the machine will be approximately $1,071.43.

A: The depreciation rate affects the value of the machine by reducing its value over time. The faster the depreciation rate, the faster the machine loses its value. In this case, the machine loses $\frac{1}{4}$ of its value every year, which means that its value decreases rapidly over time.

A: Yes, you can use this formula to calculate the value of other machines or assets that depreciate at a constant rate. Simply substitute the initial value and depreciation rate of the asset into the formula, and you will get the value of the asset after $n$ years.

A: Some factors that can affect the depreciation rate of the machine include:

Usage: The more the machine is used, the faster it will depreciate.
Maintenance: Regular maintenance can help extend the life of the machine and reduce its depreciation rate.
Inflation: Inflation can affect the value of the machine over time, and may require adjustments to the depreciation rate.
Technological advancements: Advances in technology can make the machine obsolete, reducing its value and increasing its depreciation rate.

A: This analysis can help you understand the financial implications of owning a T-shirt printing machine and make informed decisions about your business. For example, you can use this analysis to:

Determine the optimal usage of the machine: By understanding the depreciation rate of the machine, you can determine the optimal usage of the machine to maximize its value.
Plan for maintenance and repairs: By understanding the factors that affect the depreciation rate of the machine, you can plan for maintenance and repairs to extend the life of the machine.
Make decisions about investing in new equipment: By understanding the depreciation rate of the machine, you can make informed decisions about investing in new equipment or upgrading existing equipment.

In this article, we answered some frequently asked questions about the depreciation of Tessa's T-shirt printing machine. We provided formulas and examples to help you understand the depreciation rate of the machine and calculate its value after $n$ years. We also discussed some factors that can affect the depreciation rate of the machine and how you can use this analysis to make informed decisions about your business.