A maple syrup producer would like to increase production by 500 gallons per each consecutive year with a goal of producing at least 20,000 gallons of syrup cumulatively for the first five year period of production.

If the first year's production, in gallons, is given by x, write expressions for each of the other years' productions.

Year 1 = x Year 2 = Year 3 = Year 4 = Year 5 =

b) Find all possible values of syrup that could be produced the first year to meet the goal of at least 20,000 gallons cumulatively over the five-year period. You must set up a mathematical formula/rule and solve algebraically.

b) x ≥ 3000 gallons

Step-by-step explanation:

Part a)

Production of Year 1 = x gallons

By each year the producer will increase the production by 500 gallons. So,

Production in Year 2 = x + 500 gallons

Similarly,

Production in Year 3 = x + 500 + 500 = x + 1000 gallons

Production in Year 4 = x + 1000 + 500 = x + 1500 gallons

Production in Year 5 = x + 1500 + 500 = x + 2000 gallons

Part b)

The maple syrup producer wants to produced atleast 20,000 gallons in 5 years.

Total amount produced in 5 years = x + x + 500 + x + 1000 + x + 1500 + x + 2000

Total amount produced in 5 years = 5x + 5000

Since, producer wants total production to be atleast 20,000, we can set up the inequality as:

5x + 5000 ≥ 20000

Subtracting 5000 from both sides, we get:

5x ≥ 15000

Dividing both sides by 5, we get:

x ≥ 3000

This means, in the first year the production of maple syrup must be atleast 3,000 gallons i. e 3000 gallons or more to have a total of atleast 20,000 gallons in 5 years.