Mini-Mock Exam 03: Functions & Business Models
Dr. Nikolai Heinrichs & Dr. Tobias Vlćek
Name: _______________________
Reading Time: 10 minutes
Working Time: 90 minutes
Permitted Aids:
- Calculator (non-programmable without graphing capabilities)
- Drawing instruments
- No formula sheets or notes
| Grade | Points Required | Percentage |
|---|---|---|
| 1 (Excellent) | 45-50 | 90-100% |
| 2 (Very Good) | 39-45 | 77-90% |
| 3 (Good) | 32-39 | 63-77% |
| 4- (Pass) | 23-32 | 45-63% |
| 5-6 (Fail) | 0-23 | 0-45% |
Note: Passing grade requires at least 23 points (45%).
Problem 1: E-Commerce Platform Optimization [28 pts. total]
An online marketplace analyzes its pricing and demand relationships for a new product category. Market research reveals strategic information about customer behavior and cost structures.
Part A: Demand and Revenue Analysis
The demand function is linear with a maximum willingness to pay of 150 currency units (CU) when no units are sold. At a price of 30 CU, customers would purchase 40 units (Un).
Determine the linear demand function \(p(x)\) expressing price as a function of quantity. Show all steps. [4 pts.]
Show that the revenue function is given by \(R(x) = 150x - 3x^2\). Start from your demand function. [3 pts.]
Part B: Cost Structure
The company has fixed costs of 800 CU per month and variable costs that follow the function \(V(x) = 2x + 0.5x^2\) where \(x\) represents the quantity produced.
- Express the total cost function \(C(x)\) and compute the cost of producing 25 units. [4 pts.]
For verification purposes only: - \(p(x) = 150 - 3x\) - \(C(x) = 800 + 2x + 0.5x^2\)
Part C: Profit Optimization
Determine the profit function \(P(x)\) and find the quantity that maximizes profit. Use the vertex formula and verify that this is indeed a maximum. [7 pts.]
Calculate the break-even points by solving \(P(x) = 0\). Explain their significance for the business using complete sentences. [5 pts.]
Part D: Practical Constraints
- Due to warehouse limitations, the company can only stock a maximum of 20 units at any time. Determine:
- The profit at this constraint level
- The price that should be charged at this quantity
- Whether the constraint is binding (affecting the optimal solution)
Problem 2: Function Analysis and Business Application [22 pts. total]
Consider the function \(f(x) = -0.25x^2 + 4x + 5\) which models the daily profit (in hundreds of CU) of a restaurant based on the number of staff members \(x\).
Part A: Function Properties [12 pts.]
Determine the domain that makes sense in this business context. Explain your reasoning using complete sentences. [2 pts.]
Find the vertex of the function using the vertex formula \(x = -\frac{b}{2a}\). Show your calculation and interpret its meaning for the restaurant. [4 pts.]
Determine where the profit equals zero (x-intercepts). Use the quadratic formula and explain what these points represent for the business. [3 pts.]
The restaurant currently employs 12 staff members. Calculate the current profit and determine how many additional staff would optimize profit. [3 pts.]
Part B: Transformations and Composition [10 pts.]
The restaurant plans to expand to a tourist location where: - All costs increase by 20% (affecting the entire profit function) - An additional fixed cost of 300 CU per day is incurred
Write the transformed profit function \(g(x)\) for the tourist location incorporating the cost increase and additional fixed costs. Show the transformation steps. [3 pts.]
If the minimum acceptable daily profit is 500 CU (5 hundreds), determine the range of staff numbers that achieve this for:
- The original location
- The tourist location
Show your work algebraically. [4 pts.]
The company uses a staffing agency that provides workers according to the function \(w(d) = 2d + 6\), where \(d\) is the number of days in advance the request is made.
Express the profit as a composite function \((f \circ w)(d)\) for the original location and evaluate the profit when ordering 3 days in advance. [3 pts.]