A club decides to sell T-shirts for $15 as a fundraiser. It costs $20 plus $9 per T-shirt to make the T-shirts. Write and solve an equation to find how many T-shirts the club needs to make and sell in order to profit at least $150.
Let \( x \) represent the number of T-shirts the club needs to make and sell.
The revenue from selling the T-shirts would be the price per T-shirt (\$15) multiplied by the number of T-shirts (\(x\)):
Revenue = $15\(x\)
The cost of making the T-shirts includes a fixed cost of $20 plus $9 per T-shirt. So, the total cost can be expressed as:
Cost = $20 + $9\(x\)
To make a profit at least $150, the revenue needs to be at least $150 more than the cost.
Therefore, the equation to solve is:
\$15\(x\) \(\geq\) \$20 + $9\(x\) + $150
Simplifying the equation:
$15\(x\) - $9\(x\) \(\geq\) $170
$6\(x\) \(\geq\) $170
Next, divide both sides of the equation by $6 to solve for \(x\):
\(x\) \(\geq\) $170 / $6
\(x\) \(\geq\) 28.33
Since the number of T-shirts must be a whole number, the club needs to make and sell at least 29 T-shirts in order to profit at least $150.
To solve this problem, let's break it down step by step:
1. Let's first determine the total cost of making a T-shirt. We know that it costs $20 plus $9 per T-shirt. Therefore, the cost per T-shirt can be represented as $20 + $9 = $29.
2. Next, we need to determine the profit per T-shirt. Since each T-shirt is being sold for $15, the profit per T-shirt can be calculated as the selling price minus the cost price: $15 - $29 = -$14.
3. We want to find the number of T-shirts the club needs to sell in order to profit at least $150. If we assume that x represents the number of T-shirts, we can express this condition as:
Profit from x T-shirts ≥ $150
To write the equation, we need to multiply the profit per T-shirt by the number of T-shirts sold (x):
-14x ≥ 150
4. To solve for x, we need to isolate x. We can do this by dividing both sides of the inequality by -14. However, whenever we divide or multiply both sides of an inequality by a negative number, we need to flip the inequality sign:
x ≤ 150 / -14
Simplifying this expression, we get:
x ≤ -10.71
5. However, since we cannot have a negative number of T-shirts, the club cannot sell a negative fraction of a T-shirt. Therefore, we need to round down the value of x to the nearest whole number:
x ≤ -10
The club needs to make and sell at least 11 T-shirts in order to profit at least $150.