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The cost of a T-Shirt

You receive bids for purchasing hoodies for your class fundraiser from four companies. To present the pricing information, one company sent you a table, one company sent a verbal description, one company sent you an equation, and one company sent a graph. How can you compare the different representations to make a final choice?

You must select the best option for ordering the hoodies. The proposals from the four companies are listed below. The class officers also have not decided how many T-shirts should be ordered.

Part A:

1. Show all 4 offers using the same format (graphs using Desmos is recommended).

2. Compare your options and explain which company you would select for orders from 20 to 200 hoodies. (Hint: It may make sense to go with one company if you are only ordering 20 hoodies, but a different company when ordering 21-50, etc.). Explain all options from 20-200.
- Explain your reasoning clearly using the correct vocabulary terms.

Hoodies R Us: Pricing structure is sent as a table.

\begin{tabular}{|c|c|}
\hline Number of Hoodies & Price (dollars) \\
\hline 20 & 130 \\
\hline 40 & 200 \\
\hline 60 & 270 \\
\hline 80 & 340 \\
\hline 100 & 410 \\
\hline 120 & 480 \\
\hline 140 & 550 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
### Part A: Displaying the Offers in the Same Format

Let's convert all offers into tabular format for easier comparison.

#### 1. Hoodies R Us:
Hoodies R Us provided a pricing table directly, which is as follows:

[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Hoodies} & \text{Price (dollars)} \\ \hline 20 & 130 \\ \hline 40 & 200 \\ \hline 60 & 270 \\ \hline 80 & 340 \\ \hline 100 & 410 \\ \hline 120 & 480 \\ \hline 140 & 550 \\ \hline \end{array} \][/tex]

#### 2. Company B's Description:
Company B charges a base fee of \[tex]$8 plus \$[/tex]6 per hoodie.

[tex]\[ \text{Cost} = 8 + 6 \times (\text{Number of Hoodies}) \][/tex]

Calculating for 20 to 200 hoodies:

[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Hoodies} & \text{Price (dollars)} \\ \hline 20 & 128 \\ \hline 40 & 248 \\ \hline 60 & 368 \\ \hline 80 & 488 \\ \hline 100 & 608 \\ \hline 120 & 728 \\ \hline 140 & 848 \\ \hline 160 & 968 \\ \hline 180 & 1088 \\ \hline 200 & 1208 \\ \hline \end{array} \][/tex]

#### 3. Company C's Equation:
Company C uses the equation [tex]\( C = 10q + 20 \)[/tex], where [tex]\( q \)[/tex] is the number of hoodies.

Calculating for 20 to 200 hoodies:

[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Hoodies} & \text{Price (dollars)} \\ \hline 20 & 220 \\ \hline 40 & 420 \\ \hline 60 & 620 \\ \hline 80 & 820 \\ \hline 100 & 1020 \\ \hline 120 & 1220 \\ \hline 140 & 1420 \\ \hline 160 & 1620 \\ \hline 180 & 1820 \\ \hline 200 & 2020 \\ \hline \end{array} \][/tex]

#### 4. Company D's Graph Interpretation:
Company D's pricing structure appears to follow the equation [tex]\( C = 15q - 50 \)[/tex].

Calculating for 20 to 200 hoodies:

[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Hoodies} & \text{Price (dollars)} \\ \hline 20 & 250 \\ \hline 40 & 550 \\ \hline 60 & 850 \\ \hline 80 & 1150 \\ \hline 100 & 1450 \\ \hline 120 & 1750 \\ \hline 140 & 2050 \\ \hline 160 & 2350 \\ \hline 180 & 2650 \\ \hline 200 & 2950 \\ \hline \end{array} \][/tex]

### Part B: Comparing and Explaining the Choices

#### Comparison:

1. Hoodies R Us has specific rates for increments of 20 hoodies and is the most affordable option up to 140 hoodies.
2. Company B's costs increase steadily with a fixed initial charge and per-hoodie fee, making it the cheapest option for higher quantities beyond the offerings of Hoodies R Us.
3. Company C is more expensive than Company B for all quantities due to its higher per-hoodie cost and a fixed fee.
4. Company D has the steepest increase due to the highest per-hoodie cost and a fixed discount, making it the least favorable financially for nearly all quantities.

#### Analysis by Quantity:

- 20 hoodies: The best option is Hoodies R Us at \[tex]$130. - 40-140 hoodies: Continue with Hoodies R Us, which ranges from \$[/tex]200 to \[tex]$550. - 160-200 hoodies: Company B becomes beneficial at 160 hoodies costing \$[/tex]968 compared to Company C at \[tex]$1620 and Company D at \$[/tex]2350.

Overall, Hoodies R Us is the best choice when ordering up to 140 hoodies, and Company B is more cost-effective for larger orders beyond 140 hoodies.

#### Reasoning:

- Fixed Costs: Companies like Company B and Company C have fixed costs that impact initial lower quantities more significantly.
- Per-Hoodie Cost: The rate per hoodie influences overall cost heavily, which makes Company B preferable for higher quantities due to its lower per-unit cost compared to Company C and Company D.

These comparisons allow us to decide based on minimizing expenses depending on the quantity of hoodies ordered.