A tile factory earns money by charging a flat fee for delivery and a sales price of $0.25 per tile. One customer pays a total of $3,000 for 10,000 tiles. The equation $y-3,000=0.25(x-10,000)$ models the revenue of the tile factory, where $x$ is the number of tiles and $y$ is the total cost to the customer.

Which function describes the revenue of the tile factory in terms of tiles sold?
What is the flat fee for delivery?

Question

loyamanny96 loyamanny96

06-11-2026
Mathematics

Answered

A tile factory earns money by charging a flat fee for delivery and a sales price of $0.25 per tile. One customer pays a total of $3,000 for 10,000 tiles. The equation $y-3,000=0.25(x-10,000)$ models the revenue of the tile factory, where $x$ is the number of tiles and $y$ is the total cost to the customer.

Which function describes the revenue of the tile factory in terms of tiles sold?
What is the flat fee for delivery?

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T08:21:50-05:00

Start by expanding the point-slope equation $y - 3,000 = 0.25(x - 10,000)$ to find the slope-intercept form.
The resulting revenue function is $f(x) = 0.25x + 500$.
The flat fee is the constant term $b=500$, representing the cost at $x=0$.
Final answers: Revenue function is $\boxed{f(x) = 0.25x + 500}$ and the flat fee is $\boxed{\$500}$.

Explanation

Problem Analysis

The problem provides the revenue model for a tile factory in the form of a linear equation: $y - 3,000 = 0.25(x - 10,000)$, where $x$ represents the number of tiles sold and $y$ represents the total cost to the customer. We are told the factory charges a variable price per tile (represented by the slope) and a fixed, flat fee for delivery (represented by the y-intercept). We need to determine the functional form $f(x)$ for the revenue and identify the flat fee.

Deriving the Revenue Function

To find the function $f(x)$, we convert the given point-slope form equation $y - 3,000 = 0.25(x - 10,000)$ into the slope-intercept form, $y = mx + b$, where $m$ is the rate per tile and $b$ is the flat fee.
Expanding the equation:
\[y - 3,000 = 0.25x - 2,500\]
Adding 3,000 to both sides:
\[y = 0.25x - 2,500 + 3,000\]
\[y = 0.25x + 500\]
Thus, the function describing the total cost as a function of the number of tiles $x$ is $f(x) = 0.25x + 500$.

Determining the Flat Fee

The flat fee for delivery is the value of the function when no tiles are sold, i.e., when $x = 0$. From the slope-intercept form $f(x) = 0.25x + 500$, we see that:
\[f(0) = 0.25(0) + 500 = 500\]
Therefore, the flat fee for delivery is $\$500$.

Examples

Understanding linear models is essential in business for pricing strategies. For example, a catering company might charge a fixed venue rental fee plus a per-person cost for food. If a venue charges a \$1,000 booking fee and \$20 per guest, the total cost $C$ for $n$ guests can be modeled by $C(n) = 20n + 1000$. Recognizing the slope (20) and the y-intercept (1000) allows the business to predict revenue and costs for varying event sizes, helping them plan their logistics and profitability.