Answer the problems below and show or explain how you arrived at your final answer.
1. You sell premium toasters and are making a pricing decision. At a price of $100 (or “p = $100”) you predict that you can sell 30 of these premium toasters at your Scottsdale, Arizona location. If the price is raised to $110 you predict that you can sell 25 of these premium toasters at your Scottsdale, Arizona location. Find the equation of the line and you can write your answer in the following form: p = mx + b. Normally in this class, you would probably use the “y = mx + b” format. You are basically substituting the variable “y” with “p”. (20% of grade)
=(30-25)(100-110)
=5/10
=2
2. 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. The Revenue equation is “R = xp”. You are advising a local company that sells hamburgers at your state fair and have determined that the demand equation is:
p = -x + 300. Substitute this equation for “p” into the previously mentioned Revenue equation. (20% of grade)
3. A local computer company found the following Profit Equation for their new laptop:
–x2 + 66x – 800
Using “p = 0” for the breakeven point, find the breakeven values (Hint: Try the numbers “50” and “16” when factoring). (20% of grade)
4. Your toaster company’s sales were $12,000,000 in 2007 and $11,000,000 in 2009. Write your toaster company’s 2007 and 2009 sales in scientific notation. (20% of grade)
5. Using the above sales numbers in for your toaster company in 2007 and 2009 and by showing your work using scientific notation, find the percentage growth or percentage loss between 2007 and 2009. (20% of grade)
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1. To find the equation of the line representing the relationship between price (p) and quantity sold (x), we can use the formula for the equation of a straight line: y = mx + b. In this case, we substitute y with p.
We have two data points: (x1, p1) = (30, 100) and (x2, p2) = (25, 110).
First, let's find the slope (m) using the formula:
m = (p2 - p1) / (x2 - x1)
m = (110 - 100) / (25 - 30)
m = 10 / -5 = -2
Now, let's find the y-intercept (b) using one of the data points and the slope:
p = mx + b
100 = -2 * 30 + b
100 = -60 + b
b = 100 + 60
b = 160
Therefore, the equation of the line is:
p = -2x + 160
2. The demand equation is given as p = -x + 300. We can substitute this equation for p in the revenue equation R = xp.
R = x * (-x + 300)
R = -x^2 + 300x
Therefore, the revenue equation is R = -x^2 + 300x.
3. To find the breakeven values, we need to solve the equation -x^2 + 66x - 800 = 0.
We can try factoring or using the quadratic formula to solve this equation. Let's try factoring first.
When factoring, we look for two numbers that multiply to give -800 and add up to 66. Trying different pairs of numbers, we find that 50 and 16 satisfy these conditions.
Therefore, the equation can be factored as:
(-x + 50)(x - 16) = 0
Setting each factor equal to zero, we have:
-x + 50 = 0 or x - 16 = 0
Solving for x, we get:
x = 50 or x = 16
So the breakeven values are 50 and 16.
4. To write the sales numbers in scientific notation, we express them in the form a x 10^b, where a is a number between 1 and 10, and b is an integer.
Sales in 2007: $12,000,000 can be written as 1.2 x 10^7
Sales in 2009: $11,000,000 can be written as 1.1 x 10^7
5. To find the percentage growth or loss between 2007 and 2009, we can use the formula for percentage change:
Percentage change = (New value - Old value) / Old value * 100
Using the sales numbers in scientific notation, we have:
Percentage change = (1.1 x 10^7 - 1.2 x 10^7) / (1.2 x 10^7) * 100
Simplifying this expression, we get:
Percentage change = (-0.1 x 10^7) / (1.2 x 10^7) * 100
Percentage change = -0.0833 * 100
Percentage change = -8.33%
Therefore, there was a 8.33% decrease in sales between 2007 and 2009.