The student council has decided to sell t-shirts for a fundraiser. The printing company charges a $75 set-up fee plus $3 per shirt. The student council plans to sell their t-shirts for $12 per shirt. Write a function p(t) to represent the student council's profits.
P(t)= 9t – 75
Sure, I can help you with that! Here's the function p(t) to represent the student council's profits:
p(t) = (12 - 3) * t - 75
In this function, t represents the number of t-shirts sold. The expression (12 - 3) represents the profit per shirt, and multiplying it by t gives the total profit from selling t-shirts. Subtracting the set-up fee of $75 gives the final profit.
To write the function p(t) to represent the student council's profits, we need to determine the total revenue and the total cost.
Total revenue (R) is given by the selling price per shirt (12 dollars) multiplied by the number of shirts sold (t):
R = 12t
Total cost (C) comprises the set-up fee (75 dollars) plus the cost per shirt (3 dollars) multiplied by the number of shirts sold (t):
C = 75 + 3t
The profit (P) is obtained by subtracting the total cost from the total revenue:
P = R - C
= 12t - (75 + 3t)
= 12t - 75 - 3t
= 9t - 75
Therefore, the function representing the student council's profits can be written as:
p(t) = 9t - 75
To write the function p(t) to represent the student council's profits, we need to consider the costs and revenue involved in selling the t-shirts.
The costs include the set-up fee charged by the printing company and the cost per shirt. The set-up fee is a one-time cost, while the cloth cost per shirt is dependent on the number of shirts sold.
The revenue is the amount earned from selling the shirts. The revenue per shirt is the selling price.
Let's break down the components of the function:
1. The set-up fee is fixed at $75, which is the constant cost regardless of how many shirts are sold.
2. The cost per shirt is $3, and it increases linearly with the number of shirts sold. Hence, the cost due to shirts sold can be represented as $3 multiplied by the number of shirts, which is t.
3. The revenue per shirt is $12, which is the selling price of each shirt.
Now, let's put it together to form the function p(t):
p(t) = (revenue) - (costs)
The revenue can be calculated by multiplying the number of shirts sold (t) with the selling price per shirt ($12):
revenue = $12 * t
The costs consist of the set-up fee ($75) and the cost per shirt ($3 * t):
costs = $75 + ($3 * t)
Substituting the values of revenue and costs into the function, we get:
p(t) = revenue - costs
p(t) = ($12 * t) - ($75 + ($3 * t))
p(t) = $12t - $75 - $3t
p(t) = 9t - 75
Therefore, the function p(t) representing the student council's profits from selling t-shirts is p(t) = 9t - 75.