You want to sell t-shirts for $12 a piece. It cost you $40 for supplies. Is the total profit proportional to the number of t-shirts you sell? Change the scenario to be a proportional scenario.
In order to make the scenario proportional, we need to ensure that the profit is directly proportional to the number of t-shirts sold.
Let's assume that the total profit, P, is proportional to the number of t-shirts sold, T.
We can then write the equation for the total profit as:
P = kT,
where k is the constant of proportionality.
To determine if the total profit is proportional to the number of t-shirts sold, we can analyze the situation.
According to the given information, the cost of supplies is $40. Therefore, the profit from selling x t-shirts at $12 per t-shirt would be:
Profit = Revenue - Cost of Supplies = 12x - 40
However, since we want the scenario to be proportional, the profit should vary linearly with the number of t-shirts sold. This implies that the constant of proportionality, k, must be equal to $12.
Thus, the correct equation for the total profit is:
P = 12T.
In this proportional scenario, the total profit is indeed directly proportional to the number of t-shirts sold.
Show a proportional graph and equation and compare to a non proportional graph and equation.
Proportional: y =
Non Proportional: y =
Proportional: y = 12T
In a proportional scenario, the equation relating the total profit, P, to the number of t-shirts sold, T, is given by y = 12T. The constant of proportionality is 12, which represents the profit earned per t-shirt sold.
Graphically, this equation represents a straight line passing through the origin (0, 0) with a slope of 12. As the number of t-shirts sold increases, the total profit also increases linearly.
Non Proportional: y = 12T - 40
In a non proportional scenario, the equation relating the total profit, P, to the number of t-shirts sold, T, is given by y = 12T - 40. The additional term of -40 represents the cost of supplies ($40), which is subtracted from the revenue.
Graphically, this equation represents a straight line passing through the point (0, -40) with a slope of 12. The line does not pass through the origin, indicating that even when zero t-shirts are sold, there is still a cost incurred (-$40). This non-proportional scenario results in a shift downwards on the y-axis. As the number of t-shirts sold increases, the total profit still increases linearly, but at a reduced rate due to the constant subtraction of the cost of supplies.
Comparing the two scenarios, we can see that in the proportional scenario, the total profit increases directly proportional to the number of t-shirts sold, whereas in the non-proportional scenario, the total profit increases but is reduced by a fixed cost.
To determine if the total profit is proportional to the number of t-shirts sold, let's first calculate the profit for different quantities of t-shirts.
Let's say we sell x number of t-shirts. Each t-shirt is sold for $12, and the cost for supplies is $40. Therefore, the profit for selling x t-shirts can be calculated as:
Profit = (Revenue) - (Cost)
Profit = (Price per t-shirt * Number of t-shirts) - (Cost)
Profit = ($12 * x) - $40
Profit = $12x - $40
Now, let's analyze if the profit is proportional to the number of t-shirts. We can compare the profit for different quantities of t-shirts sold.
For example, if we sell 10 t-shirts:
Profit = $12 * 10 - $40
Profit = $120 - $40
Profit = $80
If we sell 20 t-shirts:
Profit = $12 * 20 - $40
Profit = $240 - $40
Profit = $200
As we can observe, the profit increases as the number of t-shirts sold increases. Therefore, in this scenario, the total profit is not proportional to the number of t-shirts sold.
If we want to change the scenario into a proportional one, we would need to adjust either the price per t-shirt or the cost of supplies so that the profit remains constant regardless of the number of t-shirts sold.
To determine if the total profit is proportional to the number of t-shirts sold, we need to examine the relationship between the profit and the number of t-shirts.
First, let's calculate the profit per t-shirt. Profit is the revenue minus the cost, so for each t-shirt sold, the profit is $12 - $40 (cost of supplies).
Profit per t-shirt = $12 - $40 = -$28
We can see that the profit per t-shirt is negative, indicating a loss. Therefore, in this scenario, the total profit is not proportional to the number of t-shirts sold.
To change the scenario to a proportional scenario, we need to ensure that the profit is positive or zero.
Let's adjust the cost of supplies. In this case, we'll assume the cost of supplies is $6 per t-shirt:
Profit per t-shirt = $12 - $6 = $6
Now, the profit per t-shirt is positive ($6), indicating a profit. In this revised scenario, the total profit will be proportional to the number of t-shirts sold.