# College Math 117 Week 2 Answer Guide- Algebra 1B -

### Week Two Algebra 1B study guide and Soultions

#### 1. In this problem, we analyze the profit found for sales of decorative tiles. A **demand ****equation **(sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).

a. Suppose a market research company finds that at a price of *p *= $20, they would sell *x *= 42 tiles each month. If they lower the price to *p *= $10, then more people would purchase the tile, and they can expect to sell *x *= 52 tiles in a month’s time. Find the equation of the line for the demand equation. Hint: Write your answer in the form p = mx + b.

Let *x *= the number of tiles sold per month. The following *t*-table shows points for the line:

*x *= quantity *p *= price sold

42 20

52 10

To get the equation for the line, use (p-p0) = m(x-x0)

m = 1

10

10

42 52

20 10 =

Using (x,p) = (42, 20), line equation is

(p-20) = -1(x-42)

p = 20 – x + 42

p = -x + 62

A company’s **revenue **is the amount of money that comes in from sales before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, *x*, by the demand equation, *p*.

b. Substitute the result you found from part a into the equation **R = xp **to find the revenue equation. Provide your answer in simplified form.

Revenue is R = xp

R = x(-x + 62) _ substituting from part a

R = -x2 + 62x _ distributing x through.

The **costs **of doing business for a company can be found by adding **fixed costs, **such as rent, insurance, wages, and **variable costs**, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $300, and the supplier’s price for a set of tile is $6. Let *x *represent the number of tile sets.

c. If *b *represents a fixed cost, what value would represent *b*?

$300

d. Find the cost equation for the tile. Write your answer in the form C = mx + b.

Let x = number of tile sets sold (quantity)

Fixed costs: 300

Variable costs: 6x

Total costs: C = 6x + 300

The **profit **made from the sale of tile sets is found by subtracting the costs from the revenue.

e. Find the Profit Equation.

P = R-C

P = -x2 + 62x – (6x + 300)

P = -x2 + 62x – 6x -300

P = -x2 + 56x – 300

f. What is the profit made from selling 20 tile sets per month?

Substitute 20 into P for *x*:

P = -202 + 56*20 -300

P = -400 + 1120 – 300

P = 420

The profit from selling 20 tile sets is $420.00.

g. What is the profit made from selling 25 tile sets each month?

Substitute 25 into P for *x*:

P = -252 + 56*25 – 300

P = -625 + 1400 – 300

P = 475

The profit from selling 25 tiles is $475.00.

h. What is the profit made from selling no tile sets each month? Interpret your answer.

Substitute 0 into *P *for *x*:

P = -02 +62*0 – 300

P = -300

There is a $300 **loss **if no tiles are sold. This is the amount of fixed costs that must be paid whether tiles are sold or not.

i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.

Substitute various values for *x *into *P *and should arrive at the following answer:

The number of tiles that will yield the highest profit is 28.

j. How much profit would you earn from the number you found in part i?

Substitute x = 28 into *P*:

P = -282 + 56*28 – 300

P = -784 + 1568 – 300

P = 484

The most profit that could be made is $484.00.

k. What price would you sell the tile sets at to realize this profit? Hint: Use the demand equation from part a.

Substitute *x *= 28 into demand equation, p = -x + 62

p = -28 + 62

p = 34

You should sell the tile sets for $34.00 each.

#### 2. The **break even **values for a profit model are the values for which you earn $0 in profit. Use the equation you created in question one to solve *P *= 0, and find your break even values.

Set *P *= 0.

-x2 + 56x – 300 = 0

x2 -56x + 300 = 0 _ multiplying by -1 to make LHS easier to factor

(x – 50) (x – 6) = 0 _ factoring LHS

x – 50 = 0 or x – 6 = 0

x = 50 or x = 6

The break even values are when you sell 6 sets of tiles or 50 sets of tiles.

#### 3. In 2002, Home Depot’s sales amounted to $58,200,000,000. In 2006, its sales were $90,800,000,000.

a. Write Home Depot’s 2002 sales and 2006 sales in scientific notation.

2002 sales: 5.82 × 1010

2006 sales: 9.08 × 1010

You can find the percent of growth in Home Depot’s sales from 2002 to 2006 by following these steps:

· Find the increase in sales from 2002 to 2006.

· Find what percent that increase is of the 2002 sales.

b. What was the percent growth in Home Depot’s sales from 2002 to 2006? Do all your work by using scientific notation.

#### 4. A customer wants to make a teepee in his backyard for his children. He plans to use lengths of PVC plumbing pipe for the supports on the teepee and he wants the teepee to be 12 feet across and 8 feet tall (see figure). How long should the pieces of PVC plumbing pipe be?

Use the Pythagorean Theorem. One leg is ½ of the 12 foot base, and the other leg is

8 feet.

c2 = 62 + 82

c2 = 36 + 64 = 100

c = 10 feet