Liam is the founder and president of the Justin Bieber Fan Club. As a fundraiser, the JBFC sells chocolate-flavored spaghetti sauce. The current price is $6.40 per jar and their average monthly sales are 250 jars. They are considering raising prices and have been surveying customers to determine if this is a good decision. They have determined that every $0.10 increase in price will cost them 5 sales per month. Determine the range of prices that they could charge in order to make at least $1500 per month. (I have tried all I can think of but I can't figure out how to solve this. Any help would be very useful. I've put it into the equation 1500=(250-5r) (6.40+0.10r)

Your equation looks good. So, just solve it and you get

(250-5r) (6.40+0.10r) >= 1500
-.5r^2 - .7r + 1600 >= 1500
.5r^2 + .7r - 100 >= 0
Now, solve the left side using the quadratic formula, and you find that

.5r^2 + .7r - 100 = 0 at
x = -22.8 and 8.8

Between those two roots, the function is positive, so since we are dealing with the real world, we need r>=0, and we have

.5r^2 + .7r + 1600 >= 1500
for 0 <= r <= 8

Now you can figure the actual price range.

I haven't learned how to find a function. Do you mind walking me through it?

Excuse me? You developed your own function. The price after r rate increases is

6.40 + 0.1r

so, for r=0, the price is 6.40
for r=8, the price is 6.40+.1*8 = 7.20

Between those two prices, she'll make at least $1500.

Take a look at

http://www.wolframalpha.com/input/?i=plot+y%3D%28250-5r%29+%286.40%2B0.10r%29%2C+y%3D1500

To solve this problem, we need to set up an equation that relates the revenue from monthly sales to the price per jar, taking into account the decrease in sales for each price increase. Let's break it down step by step:

1. First, let's define the variables:
- x = the increase in price per jar in dollars
- y = the decrease in sales per month due to the price increase

2. Next, let's determine the revenue from monthly sales. The revenue is equal to the price per jar multiplied by the number of jars sold:
revenue = (6.40 + x) * (250 - y)

3. We are given that every $0.10 increase in price will cost them 5 sales per month. This means that for every 0.10 increase in x, y will increase by 5. Therefore, we have the equation:
y = (x / 0.10) * 5

4. We want to find the range of prices that will result in at least $1500 in revenue per month. So, we need to solve the equation:
(6.40 + x) * (250 - ((x / 0.10) * 5)) ≥ 1500

Now, let's solve the equation step by step:

1. Distribute and simplify the equation:
(6.40 + x) * (250 - (5x / 0.10)) ≥ 1500
(6.40 + x) * (250 - 50x) ≥ 1500

2. Expand using the distributive property:
(1600 + 250x - 320x - 50x^2) ≥ 1500

3. Simplify and rearrange:
1600 - 70x - 50x^2 ≥ 1500
-50x^2 - 70x + 100 ≥ 0

4. Now we have a quadratic inequality. We can solve this by finding the critical points and determining where the inequality is true.
- Find the critical points by setting the equation equal to zero:
-50x^2 - 70x + 100 = 0
Use the quadratic formula to find the solutions, which are x = -0.4 and x = 2.
- Plotting these points on a number line, we have:
-∞ --- (-0.4) --- 2 --- ∞
- Test a value within each interval:
- Test x = -1: (-50(-1)^2) - 70(-1) + 100 = 60 ≥ 0, so this interval satisfies the inequality.
- Test x = 1: (-50(1)^2) - 70(1) + 100 = -20 < 0, so this interval does not satisfy the inequality.
- Test x = 3: (-50(3)^2) - 70(3) + 100 = -320 < 0, so this interval does not satisfy the inequality.

5. Therefore, the range of prices they could charge to make at least $1500 per month is:
-0.4 ≤ x ≤ 2

In conclusion, the JBFC could increase the price per jar by at least $0.10 and up to $2.00 in order to make at least $1500 per month, considering the decrease in sales due to the price increase.