Q 318 - Intermediate Algebra

Question

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for \(15 and the landscapes for \)10. She needs to sell at least $800 worth of drawings in order to earn a profit.

a) Write a system of inequalities to model this situation.

b) Graph the system.

c) Will she make a profit if she sells 20 portraits and 35 landscapes?

d) Will she make a profit if she sells 50 portraits and 20 landscapes?

Step-by-Step Solution

Answer

Part (a) A system of inequalities to model this situation is  $\left\{\begin{array}{l}x+y\ge 60\\ 15x+10y\ge 800\end{array}\right.$

Part (b) 

Part (c) No she will not make profit.

Part (d) She does make a profit

Part (a) Step 1. Form the system of inequalities.

Let $x$ represents the number of potraits and $y$ represents the number of landscapes.

Let Catlyn wants to sell 60 drawings which includes both portrait and landscapes. 

Therefore, $x+y\ge 60$

Now, she sells portraits for $\$15$ and landscapes for $\$10$. The minimum she needs to sell is $\$800$ to earn profit.

Therefore, $15x+10y\ge 800$

So, the equatons are

$\left\{\begin{array}{l}x+y\ge 60\\ 15x+10y\ge 800\end{array}\right.$

Part (b) Step 1. Graph the first inequality.

Consider the equation $x+y\ge 60$

The intercept of the graph includes $x=60,y=60$

Now test $(0,0)$ to see the position of origin.

$0\ge 60$

which is false.

Part(b) Step 2. Graph the second inequality.

Consider the equation $15x+10y\ge 800$

The intercept of the graph includes $x=53.3,y=80$.

Now test $(0,0)$ to see the position of origin.

$0\ge 800$

which is false.

Part (c) Step 1. Check the point on the inequality.

Now let's substitute the given values of portraits and landscape to look for profit.

This implies,

$$\begin{gathered} x=20,y=35 \\ 15x+10y\ge 800 \\ 15\left(20\right)+10\left(35\right)\ge 800 \\ 300+350\ge 800 \\ 650\ge 800 \end{gathered}$$

This is not true.

$$\begin{gathered} x=20,y=35 \\ x+y\ge 60 \\ 20+35\ge 60 \\ 55\ge 60 \end{gathered}$$

This is not true.

Therefore, no profit made.

Part (d) Step 1. Check the point on the inequality.

Now let's substitute the given values of portraits and landscape to look for profit.

This implies,

$$\begin{gathered} x=50,y=20 \\ 15x+10y\ge 800 \\ 15\left(50\right)+10\left(20\right)\ge 800 \\ 750+200\ge 800 \\ 950\ge 800 \end{gathered}$$

This is not true.

$$\begin{gathered} x=50,y=20 \\ x+y\ge 60 \\ 50+20\ge 60 \\ 70\ge 60 \end{gathered}$$

This is true

Therefore, She does make a profit.