Question

Angela is starting a new business, selling shirts and dresses that she sews herself. The materials for the skirts cost $13.56, and the materials for the dresses cost 18.79. She only has $80 to spend. She still has a regular job, so she is limited on the time she has to sew her products. She only wants to spend 15 hours sewing; it takes 2 hours to sew a skirt, and 4 hours to sew a dress. Answer the following questions below and show you work.1. Using the information above, write inequalities representing the constraints based on the amount she has for materials, and the amount of time she has to spend sewing.2. Graph each inequality on the same set of coordinate axes. Identify the number of skirts and dresses that could possibly maximize her profits. Round to the nearest whole number.3. Angela thinks she could sell each skirt for $30, and each dress for $40. Write an equation for the function S(x,y), the amount of money she will make in sales for x number of skirts and y number of dresses.4. Using the equation for the amount of money she could make, Identify the number of skirts and dresses that will maximize her sales, and find her total profit for selling that amount of skirts and dresses.

Answer 1

1.

Since the number of skirts is x and the number of dresses is y

Since the cost of the material of skirts is $13.56 and for the dresses is $18.79

Since she only has $80, then the 1st inequality is

$13.56x+18.79y\leq80\rightarrow(1)$

Since she takes 2 hours for one skirt and 4 hours for one dress

Since she can work only on them for 15 hours

Then the 2nd inequality is

$2x+4y\leq15\rightarrow(2)$

2.

We will graph the 2 inequalities

The red area represents inequality (1)

The blue area represents inequality (2)

The two lines red and blue intersected at the point (2.29, 2.605)

Then to get the maximum profit she has to make about 2 skirts and 3 dresses

3.

Since she will sell each skirt for $30 and $40 for each dress

The equation is

$S=30x+40y$

4.

The vertices of the common shaded area are

(0, 3.75), (2.29, 2.605), (5.9, 0)

Substitute them in the equation to find the maximum amount of profit

$\begin{gathered} S=30(0)+40(4) \\ S=\text{ \ 160} \end{gathered}$$\begin{gathered} S=2(30)+3(40) \\ S=60+120 \\ S=\text{ \ }180 \end{gathered}$$\begin{gathered} S=6(30)+0(40) \\ S=\text{ \ }180 \end{gathered}$

Since the greatest amount of selling is $177, then

The number of skirts is 6

The number of dresses is 0

The total amount of profit is

$\begin{gathered} P=160-4(18.79) \\ P=\text{ \ }84.84 \end{gathered}$$\begin{gathered} P=180-[2\times13.56+3\times18.79] \\ P=\text{ \ }96.51 \end{gathered}$$\begin{gathered} P=180-6(13.56) \\ P=\text{ \ }98.64 \end{gathered}$

The greatest profit with 6 skirts and 0 dresses

The total amount of profit is $98.64