Related Rate Problem #1 - Change in Tuition
TLDRIn this educational video, the presenter tackles a related rates problem involving tuition costs. The problem begins by identifying the current tuition and the rate at which it changes, highlighting the need for a related rates approach. With a demand equation provided (q = 440p - 0.7p^2), where q represents the number of students and p represents the tuition price, the video establishes a fixed enrollment number of 63.45 students. The rate of change in enrollment is given as 1849 students per year. Using this information, the presenter solves for the current tuition price per credit hour, which is determined to be $642.68. Subsequently, the video calculates the rate of change of the tuition price over time, resulting in a decrease of $4.02 per credit hour per year. This outcome is significant as it indicates a reduction in tuition costs, a positive development for students.
Takeaways
- 📚 The problem is a related rate problem, which involves finding how fast something is changing.
- 🔍 The given information includes a demand equation (q = 440p - 0.7p^2), where q is the number of students and p is the tuition price per credit hour.
- 📈 The current enrollment (q) is 63.45 students, which is a fixed number in this scenario.
- 🚀 The rate of change of the number of students (q) with respect to time is 18,49 students per year.
- ✅ To find the tuition price (p), substitute the given q value into the demand equation and solve the resulting quadratic equation.
- 🚫 Negative tuition prices are not feasible, so only consider the positive solution.
- 🏷 The calculated tuition price per credit hour is $642.68.
- 🔢 To determine how fast the tuition is changing, find the derivative of the demand function with respect to time.
- 📉 The rate of change of tuition (dp/dt) is negative, indicating that the tuition is decreasing.
- 💡 The tuition is decreasing by up to $4.02 per credit hour per year.
- 🎓 This decrease in tuition is beneficial for students as it makes education more accessible.
Q & A
What is the main topic of the video?
-The main topic of the video is solving a related rate problem involving the current tuition and the rate at which it is changing per credit hour.
What are the two key things we need to determine when solving a related rate problem?
-The two key things to determine are the current value (in this case, the current tuition) and the rate of change of that value (how fast the tuition is changing per credit hour).
What is the demand equation given in the video?
-The demand equation given is q = 440p - 0.7p^2, where q is the number of students and p is the tuition per credit hour.
What is the current enrollment mentioned in the video?
-The current enrollment, which represents the fixed number of students (q), is 63.45 students.
What is the rate of change of enrollment given in the video?
-The rate of change of enrollment (q with respect to time) is increasing by 1849 students per year.
How is the tuition per credit hour determined in the video?
-The tuition per credit hour is determined by solving the quadratic equation formed from the demand equation with the given current enrollment.
What is the tuition per credit hour calculated in the video?
-The tuition per credit hour is calculated to be $642.68.
Why is the negative value for tuition disregarded in the solution?
-The negative value for tuition is disregarded because it is not logical to offer a negative price per credit hour of tuition.
How is the rate of change of tuition per credit hour found in the video?
-The rate of change of tuition per credit hour is found by taking the derivative of the demand function with respect to time and solving for dp/dt.
What does the negative rate of change of tuition per credit hour indicate?
-A negative rate of change indicates that the tuition is decreasing.
By how much is the tuition decreasing per credit hour per year?
-The tuition is decreasing by $4.02 per credit hour per year.
What does the rate of change of tuition tell us about the future trend of tuition costs?
-The rate of change of tuition tells us that the tuition costs are expected to decrease by $4.02 per credit hour per year, indicating a trend towards lower tuition fees.
Outlines
📚 Introduction to Related Rate Problem
The video begins with an introduction to a related rate problem, emphasizing the need to identify what is being sought: the current tuition and the rate at which it changes. It establishes that the problem involves a demand equation (q = 440p - 0.7p^2), with q representing the number of students and p representing the tuition price. The video provides a fixed number (current enrollment at 63.45 students) and a rate of change (increase of 1849 students per year). The first objective is to find the tuition price (p), which is done by substituting the given values into the demand equation and solving the resulting quadratic equation. The tuition price is found to be $642.68 per credit hour, disregarding the negative solution as it's not logical for pricing.
📉 Calculating the Rate of Change of Tuition
After determining the tuition price, the video moves on to calculate how fast the tuition is changing. This involves finding the derivative of the demand function with respect to time to get the rate of change of p (dp/dt). The derivative of q with respect to time (dq/dt) is established as 1849, and the derivative of the demand function is manipulated to isolate dp/dt. By substituting the known values of dq/dt and p into the equation, the rate of change of tuition (dp/dt) is calculated to be -4.02, indicating that the tuition is decreasing by $4.02 per credit hour per year. This suggests a positive development as the cost of tuition is going down.
Mindmap
Keywords
💡Related Rate Problem
💡Tuition
💡Demand Equation
💡Rate of Change
💡Quadratic Function
💡Derivative
💡Quadratic Formula
💡Graphing
💡Negative Price
💡Decreasing Tuition
💡Per Credit Hour
Highlights
The video presents an example of a related rate problem, focusing on determining the current tuition and how fast it's changing.
The demand equation is introduced as q = 440p - 0.7p^2, where q represents the number of students and p represents the tuition price.
A fixed number, current enrollment at 63.45 students, is given to establish a base for the related rate problem.
The rate of change of the number of students (q) is provided as 1849 students per year, indicating an increasing trend.
The goal is to find the tuition price (p) using the given demand equation and current enrollment.
The demand equation is rearranged to solve for p, resulting in a quadratic function.
The solutions for p are found to be -1410 and 642.6, but the negative price is logically disregarded.
The tuition price is determined to be $642.68 per credit hour.
The next step is to find out how fast the tuition is changing, which requires finding the derivative of the demand function.
The derivative of the demand function with respect to time is calculated to be dq/dt = 440dp/dt - 1.4pdp/dt.
The derivative is solved for dp/dt, resulting in dp/dt = dq/dt / (440 - 1.4p).
The rate of change of tuition (dp/dt) is calculated using the given rate of change of students and the calculated tuition price.
The calculated dp/dt is found to be negative, indicating that the tuition is decreasing.
The tuition is decreasing by up to $4.02 per credit hour per year.
The conclusion is that the tuition price is not only determined but also shown to be on a decreasing trend.
The video provides a clear example of applying related rate problems to real-world scenarios, such as tuition pricing.
The mathematical process of solving for the tuition price and its rate of change is demonstrated step by step.
The use of the quadratic formula and derivatives are highlighted as key mathematical tools in solving the problem.
Transcripts
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