joanne sells silk - screened t - shirts at community festivals and craft fairs. her marginal cost to produce…

joanne sells silk - screened t - shirts at community festivals and craft fairs. her marginal cost to produce one t - shirt is $3.50. her total cost to produce 50 t - shirts is $225, and she sells them for $9 each. a. find the linear cost function for joannes t - shirt production. b. how many t - shirts must she produce and sell in order to break even? c. how many t - shirts must she produce and sell to make a profit of $600? a. the linear cost function is c(x)=. b. joanne must produce and sell t - shirts in order to break even, because the number of t - shirts must be a whole number. c. joanne must produce and sell t - shirts to make a profit of $600, because the number of t - shirts must be a whole number.

joanne sells silk - screened t - shirts at community festivals and craft fairs. her marginal cost to produce one t - shirt is $3.50. her total cost to produce 50 t - shirts is $225, and she sells them for $9 each. a. find the linear cost function for joannes t - shirt production. b. how many t - shirts must she produce and sell in order to break even? c. how many t - shirts must she produce and sell to make a profit of $600? a. the linear cost function is c(x)=. b. joanne must produce and sell t - shirts in order to break even, because the number of t - shirts must be a whole number. c. joanne must produce and sell t - shirts to make a profit of $600, because the number of t - shirts must be a whole number.

Answer

Explanation:

Step1: Find the fixed - cost

The marginal cost $m = 3.50$. The total - cost formula is $C(x)=mx + b$, where $x$ is the number of T - shirts, $m$ is the marginal cost, and $b$ is the fixed cost. When $x = 50$, $C(50)=225$. Substitute into the formula: $225=3.5\times50 + b$. $225 = 175 + b$. Solve for $b$: $b=225 - 175=50$.

Step2: Write the linear cost function

The linear cost function $C(x)$ with $m = 3.5$ and $b = 50$ is $C(x)=3.5x + 50$.

Step3: Find the break - even point

The revenue function $R(x)$ is $R(x)=9x$ (since each T - shirt is sold for $9$). At the break - even point, $R(x)=C(x)$. So, $9x=3.5x + 50$. Subtract $3.5x$ from both sides: $9x-3.5x=50$, $5.5x = 50$, $x=\frac{50}{5.5}=\frac{500}{55}=\frac{100}{11}\approx9.09$. Since the number of T - shirts must be a whole number, we round up to $x = 10$.

Step4: Find the number of T - shirts for a profit of $600$

The profit function $P(x)=R(x)-C(x)$. We want $P(x)=600$, so $9x-(3.5x + 50)=600$. Expand the left - hand side: $9x-3.5x-50 = 600$. Combine like terms: $5.5x-50 = 600$. Add $50$ to both sides: $5.5x=600 + 50=650$. Solve for $x$: $x=\frac{650}{5.5}=\frac{6500}{55}=\frac{1300}{11}\approx118.18$. Since the number of T - shirts must be a whole number, we round up to $x = 119$.

Answer:

a. $C(x)=3.5x + 50$ b. $10$ c. $119$