Piecewise Functions Word Problems Quiz

Let's solve both problems step by step.

---

Problem 1: T-Shirt Purchase

Given:
- Custom Ink sells up to 30 shirts at $12 per shirt.
- If more than 30 shirts are bought, the remaining shirts (beyond 30) cost $10 each.
- Mr. Scardina has 143 students, so he needs 143 shirts.

We are asked to:
1. Write a piecewise function for the individual cost per shirt as a function of the number of shirts made.
2. Calculate how much he will have to pay for 143 shirts.

---

#### Step 1: Define the Piecewise Function

Let $ x $ = number of shirts ordered.

We are modeling the cost per shirt, not the total cost.

But note: The individual cost per shirt depends on whether it's part of the first 30 or beyond.

So:

- For $ x \leq 30 $: Each shirt costs $12.
- For $ x > 30 $: The first 30 shirts are $12 each, and each additional shirt (i.e., from 31 onward) is $10.

But the question asks for the individual cost of a t-shirt as a function of the number of shirts made.

This means: if you order $ x $ shirts, what is the price per shirt?

Wait — here's a key point: The individual cost per shirt is not constant when ordering more than 30, but only the price for extra shirts changes.

However, the cost per shirt in the context of the total cost might be interpreted as average cost per shirt or price paid per shirt depending on how many were ordered.

But the problem says: "individual cost of a t-shirt as a function of the number of shirts made"

This suggests that each shirt's cost depends on how many are being made.

But actually, the pricing structure is:
- First 30 shirts: $12 each
- Any additional shirts (beyond 30): $10 each

So the individual cost for a given shirt is:
- $12, if it's one of the first 30
- $10, if it's beyond 30

But since we're writing a function based on the number of shirts made, we can define the cost per shirt as:

> But this is ambiguous: does "individual cost" mean the price paid for that shirt, or the average cost per shirt?

Let’s re-read:
> "Write a piecewise function to represent individual cost of a t-shirt as a function of the number of shirts made."

Hmm — “individual cost” likely means how much each shirt costs, which depends on whether it's among the first 30 or not.

But since we’re expressing it as a function of x (number of shirts), we need to think carefully.

Actually, the cost per shirt is not a function of x in the sense of changing per shirt — rather, the price per shirt is fixed based on position in the order.

But perhaps they want the average cost per shirt? Or maybe they want the price paid per shirt, where:
- For orders ≤ 30: all shirts are $12
- For orders > 30: first 30 are $12, rest are $10

But again, individual cost could mean the price of each shirt.

But since all shirts are identical, the individual cost isn't really a function of total quantity unless we interpret it as:

> What is the price per shirt when ordering $ x $ shirts?

But that’s not consistent — because some shirts are $12, others $10.

So perhaps the intended meaning is:

> Let $ C(x) $ = total cost for $ x $ shirts.

Then the individual cost would be $ C(x)/x $, the average cost per shirt.

But the problem says "individual cost of a t-shirt", which may mean price per shirt, but in bulk pricing, it's usually average cost.

Alternatively, maybe they just want the total cost function, and then compute the total.

But let's look at the wording again:

> "Write a piecewise function to represent individual cost of a t-shirt as a function of the number of shirts made."

This is confusing. Because if you make 50 shirts, the individual cost for each shirt is not a single value — some are $12, some $10.

So perhaps the intended interpretation is:

> The cost per shirt depends on the quantity ordered.

But that doesn’t make sense either — unless they mean what price per shirt applies based on the order size.

Ah! Here's a better interpretation:

They probably want the price per shirt that applies when ordering $ x $ shirts.

But that’s still ambiguous.

Wait — maybe they mean: What is the total cost, and then divide by $ x $ to get average cost per shirt?

But the word "individual" suggests per shirt.

Let’s assume the intended meaning is:

> Define a function $ c(x) $ = cost per shirt when ordering $ x $ shirts, where:
> - If $ x \leq 30 $, then each shirt costs $12 → $ c(x) = 12 $
> - If $ x > 30 $, then the first 30 are $12, and the rest are $10 → but now, the individual cost varies.

So you cannot assign a single cost per shirt for $ x > 30.

Therefore, the only logical interpretation is:

> They want the total cost function $ C(x) $, and then use it to find the total for 143 shirts.

And perhaps the phrase “individual cost” is a miswording — it should be total cost.

Alternatively, maybe they want the average cost per shirt.

Let’s check common interpretations.

In most such problems, the goal is to write the total cost as a piecewise function.

Let’s go with that.

---

#### ✔ Correct Interpretation: Total Cost Function

Let $ C(x) $ = total cost for $ x $ shirts.

Then:

- If $ x \leq 30 $: $ C(x) = 12x $
- If $ x > 30 $: $ C(x) = 12 \times 30 + 10 \times (x - 30) = 360 + 10(x - 30) $

Simplify:
$$
C(x) =
\begin{cases}
12x & \text{if } x \leq 30 \\
360 + 10(x - 30) & \text{if } x > 30
\end{cases}
$$

Now simplify the second case:
$$
C(x) = 360 + 10x - 300 = 10x + 60 \quad \text{for } x > 30
$$

So:
$$
C(x) =
\begin{cases}
12x & \text{if } x \leq 30 \\
10x + 60 & \text{if } x > 30
\end{cases}
$$

Now, Mr. Scardina wants 143 shirts → $ x = 143 > 30 $

So:
$$
C(143) = 10(143) + 60 = 1430 + 60 = \$1490
$$

✔ So he will pay $1490.

---

But wait — the question says “individual cost”. So let's address that.

If “individual cost” means average cost per shirt, then:

$$
\text{Average cost per shirt} = \frac{C(x)}{x}
$$

For $ x = 143 $:
$$
\frac{1490}{143} \approx \$10.42 \text{ per shirt}
$$

But the problem says “individual cost of a t-shirt”, which likely refers to price paid per shirt, but since prices vary, we must conclude that:

> The individual cost is not constant — it’s $12 for first 30, $10 for others.

But since the function is defined as a function of the number of shirts made, it's impossible to give a single value.

Hence, the most reasonable interpretation is that they want the total cost function, and the term “individual” is misleading.

Alternatively, maybe they want:

> $ c(x) $ = cost per shirt for an order of size $ x $

But again, that’s not well-defined.

So I believe the intended answer is:

---

✔ Final Answer for Problem 1:

Piecewise function for total cost:

$$
C(x) =
\begin{cases}
12x & \text{if } x \leq 30 \\
10x + 60 & \text{if } x > 30
\end{cases}
$$

For $ x = 143 $:
$$
C(143) = 10(143) + 60 = 1430 + 60 = \boxed{\$1490}
$$

So Mr. Scardina will pay $1490.

---

Problem 2: Mountain Bike Rental

Given:
- Flat fee of $25 for the first three hours
- $5 for each additional hour

Find:
1. A piecewise function for the cost of renting for $ h $ hours
2. How much to rent for 7 hours

---

Let $ C(h) $ = cost for $ h $ hours

Break into cases:

- If $ h \leq 3 $: cost = $25
- If $ h > 3 $: cost = $25 + 5 \times (h - 3)

So:

$$
C(h) =
\begin{cases}
25 & \text{if } h \leq 3 \\
25 + 5(h - 3) & \text{if } h > 3
\end{cases}
$$

Simplify the second case:
$$
C(h) = 25 + 5h - 15 = 5h + 10 \quad \text{for } h > 3
$$

So:
$$
C(h) =
\begin{cases}
25 & \text{if } h \leq 3 \\
5h + 10 & \text{if } h > 3
\end{cases}
$$

Now, for $ h = 7 $:
$$
C(7) = 5(7) + 10 = 35 + 10 = \boxed{\$45}
$$

---

✔ Final Answer for Problem 2:

Piecewise function:
$$
C(h) =
\begin{cases}
25 & \text{if } h \leq 3 \\
5h + 10 & \text{if } h > 3
\end{cases}
$$

Cost for 7 hours: $ \boxed{45} $

---

🟩 Summary:

Problem 1:
- Piecewise total cost function:
$$
C(x) =
\begin{cases}
12x & x \leq 30 \\
10x + 60 & x > 30
\end{cases}
$$
- For 143 shirts: $ C(143) = 10(143) + 60 = \boxed{1490} $

Problem 2:
- Piecewise rental cost:
$$
C(h) =
\begin{cases}
25 & h \leq 3 \\
5h + 10 & h > 3
\end{cases}
$$
- For 7 hours: $ C(7) = 5(7) + 10 = \boxed{45} $

Let me know if you'd like graphs or further explanation!