A 135-kg steer gains 3.5kg/day and costs 80 cents/day to keep. The market price for beef cattle is $1.65/kg, but the price falls by 1 cent/day. When should the steer be sold to maximize profit?
Revenue= (Price/units)(#of units)
Cost=(cost/unit)(# of units) + set costs
Profit= Revenue-Expenses.
show the revenue and costs.
put in vertex form.
x=days
135kg +3.5kg/day*x =weight
SalePrice= $(1.65-0.01*x)/kg*weight
cost= $0.8/day*x
profit=Saleprice-cost
profit(x)=(1.65-0.01x)(135 +3.5x)-0.8x
profit(x)= 1.65*135+1.65*3.5x-0.01*135x-0.01*3.5x^2-0.8x
= 222.750+4.4250x-0.035x^2-0.8x
= 222.750+3.625x-0.035x^2
set the derivative to 0 to find max profit
dprofit(x)/dx=0
3.625-0.07x=0
x=3.625/0.07
x=51.7857~=52
hope this helps
To determine when the steer should be sold to maximize profit, we need to calculate the revenue and costs at different time intervals.
Let's start by calculating the revenue at a given time interval. The revenue is calculated by multiplying the market price per kilogram by the weight of the steer. Since the market price falls by 1 cent/day, we need to account for this decrease as well.
Revenue = (Market Price - Price Decrease/units) * (#of units)
As per the question, the initial weight of the steer is 135 kg, and it gains 3.5 kg/day. So the weight of the steer at any given time t is 135 + 3.5t kg.
Now, let's calculate the revenue in terms of the weight of the steer:
Revenue(t) = (Market Price - Price Decrease/units) * (Weight of steer at time t)
Next, let's calculate the cost at a given time interval. The cost is determined by multiplying the cost per day to keep the steer by the number of days the steer has been kept, and adding any fixed costs.
Cost = (Cost per day * # of days) + Fixed Costs
According to the question, the cost to keep the steer is 80 cents/day. Therefore, the cost at any given time t is:
Cost(t) = (80 cents/day * t days) + Fixed Costs
Now that we have the revenue and cost functions, we can calculate the profit function by subtracting the cost from the revenue:
Profit(t) = Revenue(t) - Cost(t)
To find the time at which the profit is maximized, we need to convert the profit function into vertex form, which is in the form: y = a(x-h)^2 + k
To find the vertex form, we need to rewrite the profit function in the form: Profit(t) = a(t-h)^2 + k
Now, we can analyze the vertex form to determine the values of a, h, and k.
The vertex form represents a parabola, where h represents the x-coordinate of the vertex, and k represents the y-coordinate of the vertex. In our case, the vertex represents the time at which the profit is maximized.
Once we find the values of a, h, and k, we can determine the time (t) at which the profit is maximized by taking the value of h.
Note that the fixed costs are not provided in the question, so we cannot provide a specific calculation for them. However, you can substitute the given values into the formulas provided to find the optimal time to maximize profit once the fixed costs are known.