You own a small winery that produces two types of Chardonnay, an extra extra oaky one and an oaky one. The wines are produced from the same grapes. Furthermore, both wines are aged in oak barrels, and have additional oak chips added into the oak barrels. However, the extra extra oaky one requires twice as many added oak chips as the oaky one. The extra extra oaky Chardonnay has a profit of $15 per bottle, and the oaky Chardonnay has a profit of $10 per bottle. You have enough grapes to make a total of 500 bottles of Chardonnay (regardless of if the type is extra extra oaky or oaky), and enough oak chips to make a total of 800 bottles of oaky Chardonnay. How many bottles of oaky Chardonnay should you produce?

Question

You own a small winery that produces two types of Chardonnay, an extra extra oaky one and an oaky one. The wines are produced from the same grapes. Furthermore, both wines are aged in oak barrels, and have additional oak chips added into the oak barrels. However, the extra extra oaky one requires twice as many added oak chips as the oaky one. The extra extra oaky Chardonnay has a profit of $15 per bottle, and the oaky Chardonnay has a profit of $10 per bottle. You have enough grapes to make a total of 500 bottles of Chardonnay (regardless of if the type is extra extra oaky or oaky), and enough oak chips to make a total of 800 bottles of oaky Chardonnay. How many bottles of oaky Chardonnay should you produce?

Answer 1

let number of oaky bottles be x

let number of extra oaky bottles be y

x+y ≤ 500 based on grapes
2x + y ≤ 800 based on chips

graph both lines and consider the region bounded by the x-axis, the y-axis and the two boundary lines
It can be easily found that the two boundary lines intersect at (300,200)
Also considering the intercepts, the "corners" of the region are
(0,0), (0,250), (300,200) and (400,0)

The profit equation would be
P = 10x + 15y
for (300,200), P = 3000 + 3000 = 6000
for (0,250) , P = 0 + 3750
for (400,0) , P = 4000 + 0 = 4000

They should produce 300 bottles of oaky, and 200 of the extra oaky.

Answer 2

To find out how many bottles of oaky Chardonnay you should produce, we need to consider the availability of grapes and oak chips, as well as the profit margin for each type of wine.

Let's start by determining the maximum number of bottles of extra extra oaky Chardonnay you can produce. Since you have enough grapes for a total of 500 bottles regardless of the type, we can allocate a portion of those grapes to the extra extra oaky variety.

Assuming x represents the number of bottles of extra extra oaky Chardonnay, we can set up the following equation:
x + (x/2) ≤ 500

This equation represents the total number of bottles of extra extra oaky Chardonnay being less than or equal to 500. The x/2 term represents the number of bottles of grapes needed for the oaky Chardonnay.

Simplifying the equation:
(x + x/2) ≤ 500
(3x/2) ≤ 500
3x ≤ 1000
x ≤ 333.33

Since you cannot produce a fraction of a bottle, the maximum number of bottles for the extra extra oaky Chardonnay is 333.

Now, let's determine the maximum number of bottles of oaky Chardonnay. We have enough oak chips for a total of 800 bottles of oaky Chardonnay. Since the extra extra oaky Chardonnay requires twice as many added oak chips, the oak chip demand per bottle for oaky Chardonnay is m/2, where m is the number of bottles.

To find the maximum number of bottles for oaky Chardonnay, we can set up the following inequality:
m/2 ≤ 800

Simplifying the inequality:
m ≤ 1600

Since you cannot produce more than 500 bottles in total, the maximum number of bottles for oaky Chardonnay is 500.

To optimize your profit, you should produce the maximum number of bottles for oaky Chardonnay, which is 500 bottles. This will ensure efficient use of the available oak chips while still utilizing the maximum number of grapes.