Math Problem Statement
Solution
Let's break down both questions from the image step by step.
Problem 1: Cost Function Minimization
Given:
- Total cost function
You need to find:
- The number of units that minimizes the average cost per unit.
- The minimum average cost.
Step 1: Average Cost Function
The average cost function is the total cost divided by the number of units :
Simplifying this expression:
Step 2: Minimizing Average Cost
To find the number of units that minimizes the average cost, we need to differentiate the average cost function and set the derivative equal to zero:
Set this equal to zero:
Thus, the number of units that minimizes the average cost is .
Step 3: Minimum Average Cost
Now, substitute back into the average cost function:
The minimum average cost is $47 per unit.
Problem 2: Maximizing Profit for the Apartment
Given:
- 50 units in total.
- Rent of $740 fills all units, but 1 unit becomes vacant for each $20 increase in rent.
- Monthly cost to maintain the building is $12 per unit.
You need to find the rent that maximizes profit.
Step 1: Revenue Function
Let the number of vacant units be . For every $20 increase in rent, 1 unit becomes vacant, so the rent is . If there are vacant units, the number of rented units is . The revenue function is:
Expand this expression:
Step 2: Cost Function
The cost to maintain the building is $12 per rented unit. Therefore, the cost function is:
Step 3: Profit Function
The profit function is revenue minus cost:
Step 4: Maximizing Profit
To find the maximum profit, differentiate the profit function and set the derivative equal to zero:
Set this equal to zero:
Since must be an integer, round to the nearest integer, .
Step 5: Calculate Optimal Rent
Now, substitute into the rent formula:
The optimal rent to maximize profit is $880.
Would you like more details or have any questions about these solutions?
Here are some related questions:
- How do you find the derivative of a function using the power rule?
- What are the conditions to check for a maximum or minimum value in a function?
- Can you explain how revenue functions and cost functions combine to form profit functions?
- How do second-order derivatives help in identifying whether a critical point is a maximum or minimum?
- What happens to the average cost function as production increases indefinitely?
Tip: When optimizing costs or profits, always clearly define the objective function (average cost, profit, etc.) before proceeding with differentiation.
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Math Problem Analysis
Mathematical Concepts
Cost minimization
Profit maximization
Differentiation
Revenue function
Quadratic functions
Formulas
C(x) = 360 + 35x + 0.1x^2
AC(x) = C(x) / x
Revenue function: R(v) = (50 - v)(740 + 20v)
Profit function: P(v) = Revenue - Cost
Theorems
Derivative of cost function to find minimum cost
Derivative of profit function to find maximum profit
Suitable Grade Level
Undergraduate or Advanced High School
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