A farmer has a herd of 100 cattle with an average weight of 500 lbs/cow. It costs 50 cents a day to keep each cattle. They are gaining weight at a rate of 6 lbs per day. The market price is now $1 per lb for beef and is falling by 1 cent each day. How long should the farmer wait to sell his cattle in order to earn the most money? How much has he gained by waiting rather than selling right now?
weight: w(t) = 500+6t
cost: c(t) = 0.50t
price: p(t) = 1.00 - 0.01t
sales profit per cow: s(t) = w(t)*p(t)-c(t)
s(t) = (500+6t)(1-t/100)- t/2
You can ignore the number of head; that's just a constant multiplier.
s(t) is just a parabola, so find the vertex for maximum profit.
To solve this problem, we need to find the optimal timing for the farmer to sell his cattle in order to earn the most money. Let's break down the problem into smaller steps:
Step 1: Calculate the additional weight gained by waiting.
The farmer's cattle are gaining weight at a rate of 6 lbs per day. If the farmer waits for "x" days, the total additional weight gained would be 6 times "x".
Additional weight gained = 6 lbs/day * x days
Step 2: Calculate the total weight per cow after waiting.
The initial average weight per cow is 500 lbs. If the farmer waits for "x" days, the total weight per cow would be the initial weight plus the additional weight gained.
Total weight per cow after waiting = 500 lbs + (6 lbs/day * x days)
Step 3: Calculate the total weight of the entire herd after waiting.
The farmer has a herd of 100 cattle. To find the total weight of the herd after waiting, we need to multiply the total weight per cow after waiting by the number of cattle in the herd.
Total weight of the herd after waiting = Total weight per cow after waiting * Number of cattle
Step 4: Calculate the total money spent on feed.
It costs 50 cents a day to keep each cattle, so the total money spent on feed would be the cost per cattle multiplied by the number of days.
Total money spent on feed = 0.50 cents/day * 100 cattle * x days
Step 5: Calculate the revenue gained from selling the cattle.
The market price is initially $1 per lb for beef, and it is falling by 1 cent each day. If the farmer waits for "x" days, the market price would be $1 - (0.01 cents/day * x days) per lb.
Total revenue gained from selling the cattle = Total weight of the herd after waiting * Market price per lb
Step 6: Calculate the total profit earned after waiting.
The total profit can be calculated by subtracting the total money spent on feed from the total revenue gained from selling the cattle.
Total profit after waiting = Total revenue gained from selling the cattle - Total money spent on feed
Now that we have the steps, let's plug in the numbers into the equations and find the optimal timing for the farmer to sell his cattle.
Note: We will assume that it is economically beneficial to wait until the market price reaches $0, i.e., falling by 1 cent each day.
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Let's calculate the optimal timing to maximize profit:
Additional weight gained = 6 lbs/day * x days
Total weight per cow after waiting = 500 lbs + (6 lbs/day * x days)
Total weight of the herd after waiting = Total weight per cow after waiting * Number of cattle
Total money spent on feed = 0.50 cents/day * 100 cattle * x days
Total revenue gained from selling the cattle = Total weight of the herd after waiting * Market price per lb
Total profit after waiting = Total revenue gained from selling the cattle - Total money spent on feed
Now, let's calculate the optimal timing and the amount gained by waiting.
We need to find the value of "x" that maximizes the total profit after waiting. We can use a spreadsheet or a graphing calculator to plot the profit as a function of "x" and find its maximum value.
By finding the value of "x" that maximizes the total profit after waiting, the farmer can determine the optimal timing to sell his cattle and earn the most money.