Podcast
Questions and Answers
For the profit function $P(t) = t^3 - 9t^2 + 40t + 50$ over 8 years, what initial analysis helps determine if cost-cutting measures are effective?
For the profit function $P(t) = t^3 - 9t^2 + 40t + 50$ over 8 years, what initial analysis helps determine if cost-cutting measures are effective?
- Graphing $P(t)$ to observe its trend and identify intervals of increasing profit. (correct)
- Calculating the average profit $P(t)/8$; a decreasing value indicates problems.
- Finding when $P(t) = 0$ to determine break-even points; fewer points mean effectiveness.
- Calculating $P(0)$ to find the initial profit; a negative value indicates ineffectiveness.
A retailer models revenue as $R(q) = -0.003q^3 + 1.35q^2 + 2q + 8000$ where $q$ is the amount spent on advertising (in thousands). What does the 'point of diminishing returns' signify in this model?
A retailer models revenue as $R(q) = -0.003q^3 + 1.35q^2 + 2q + 8000$ where $q$ is the amount spent on advertising (in thousands). What does the 'point of diminishing returns' signify in this model?
- The point where the cost of advertisement equals the revenue generated.
- The advertising expense ($q$) at which the total revenue ($R(q)$) is maximized.
- The advertising expense ($q$) at which the rate of revenue increase begins to decrease. (correct)
- The advertising expense ($q$) at which revenue begins to decrease.
Given the revenue function $R(q) = -0.003q^3 + 1.35q^2 + 2q + 8000$, how would one determine if it's worth investing more than $150,000 on advertisement?
Given the revenue function $R(q) = -0.003q^3 + 1.35q^2 + 2q + 8000$, how would one determine if it's worth investing more than $150,000 on advertisement?
- Compare $R'(150)$ to the cost of advertisement to see if further investment has a positive rate of return (correct)
- Check if $R(150) > 150,000$ to ensure the return exceeds the investment.
- Compare $R(150)$ with the maximum value of $R(q)$ to see if further investment is justified.
- Calculate the total cost of advertisement minus $R(150)$ to determine profitability.
If setting the price of an item at $10 results in no sales, but 500 items can be sold for each dollar below $10, with fixed costs of $3000 and a marginal cost of $2 per item, which expression represents the profit function?
If setting the price of an item at $10 results in no sales, but 500 items can be sold for each dollar below $10, with fixed costs of $3000 and a marginal cost of $2 per item, which expression represents the profit function?
A company's monthly profit is modeled by $P(q) = -5q^2 + 1300q - 15000$, where $q$ is the number of items sold. At what production level, $q$, is profit maximised?
A company's monthly profit is modeled by $P(q) = -5q^2 + 1300q - 15000$, where $q$ is the number of items sold. At what production level, $q$, is profit maximised?
Why would a manufacturer need to set $P'(q) = 0$ for the daily profit function $P(q) = -0.2q^3 + 2q^2 - 1000$?
Why would a manufacturer need to set $P'(q) = 0$ for the daily profit function $P(q) = -0.2q^3 + 2q^2 - 1000$?
For a manufacturer with total cost $C(q) = 400 + 4q + 0.0001q^2$ and selling price $p = 10 - 0.0004q$, which expression represents the profit function?
For a manufacturer with total cost $C(q) = 400 + 4q + 0.0001q^2$ and selling price $p = 10 - 0.0004q$, which expression represents the profit function?
A box with a square base and no top must enclose 100 $m^3$. What equation must be solved to minimize the total material used?
A box with a square base and no top must enclose 100 $m^3$. What equation must be solved to minimize the total material used?
Given $p = \sqrt{300 - 0.5q}$, where $p$ is the unit price and $q$ is units manufactured, what step is next in maximizing weekly revenue?
Given $p = \sqrt{300 - 0.5q}$, where $p$ is the unit price and $q$ is units manufactured, what step is next in maximizing weekly revenue?
A car rental company models the number of cars rented per day as $N(p) = 1000 - 5p$, where $p$ is the rental price. To maximize revenue, what function should be analyzed?
A car rental company models the number of cars rented per day as $N(p) = 1000 - 5p$, where $p$ is the rental price. To maximize revenue, what function should be analyzed?
Flashcards
Profit Function P(t)
Profit Function P(t)
A function to model profit over the next 8 years, used to evaluate cost-cutting measures.
Revenue Function R(q)
Revenue Function R(q)
A function to model revenue based on money spent on advertising
Point of Diminishing Returns
Point of Diminishing Returns
The point at which the rate of increase in revenue decreases with each additional dollar of investment.
Monthly Profit Function P(q)
Monthly Profit Function P(q)
Signup and view all the flashcards
Total Cost Function C(q)
Total Cost Function C(q)
Signup and view all the flashcards
Price Function p(q)
Price Function p(q)
Signup and view all the flashcards
Customer number Function N(p)
Customer number Function N(p)
Signup and view all the flashcards
Study Notes
- The document comprises optimization problems for business and management, including profit maximization, cost minimization, and revenue optimization.
Cost-Cutting Measures Evaluation
- A company is evaluating cost-cutting measures, using a profit function P(t) = t³ - 9t² + 40t + 50 for the next 8 years (0 < t < 8), with a request to check the effectiveness of these measures graphically.
Revenue Optimization
- A retailer models revenue R(q) as a function of advertisement spending q (in thousands): R(q) = -0.003q³ + 1.35q² + 2q + 8000, where 0 ≤ q ≤ 400.
- Questions include identifying the point(s) of diminishing returns and assessing if investing more than $150,000 on advertisement is worthwhile.
Profit Maximization
- If an item is priced at $10, it cannot be sold, but 500 items can be sold for each dollar below $10. Given fixed costs of $3000 and a marginal cost of $2 per item, find the maximum profit.
- For a manufacturing company, the monthly profit for q sold items is P(q) = -5q² + 1300q - 15000, and the goal is to find the maximum monthly profit.
- A manufacturer's daily profit is P(q) = -0.2q³ + 2q² - 1000, with q ≥ 0, and the objective is to find the maximum monthly profit graphically.
Production Level Optimization
- The total cost per unit sold is C(q) = 400 + 4q + 0.0001q², and each unit is sold at p = 10 - 0.0004q; find the production level that maximizes profit.
Box Dimensions Optimization
- A box with a square base and no top needs to enclose 100m³ and the goal is to determine dimensions that minimize total material used.
Rectangle Dimension Optimization
- Find dimensions of a rectangle with a maximum area and fixed perimeter P = 100.
Revenue Maximization with Price-Demand
- Given p = √(300 - 0.5q), where p is the unit price and q is the number of units manufactured per week, determine how many units should be manufactured and sold to maximize revenue.
Car Rental Revenue Optimization
- A car rental company models car rentals n per day as N(p) = 1000 - 5p, where p is the price per day (50 ≤ p ≤ 200).
- They rent all cars at $50/day or less and none at $200/day or more; find the price to maximize revenue, assuming prices between $50 and $200.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
