Related Rate Problem #1 - Change in Tuition

Sun Surfer Math
31 Mar 202207:59
EducationalLearning
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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
  • πŸ“š The problem is a related rate problem, which involves finding how fast something is changing.
  • πŸ” 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.
  • πŸ“ˆ The current enrollment (q) is 63.45 students, which is a fixed number in this scenario.
  • πŸš€ The rate of change of the number of students (q) with respect to time is 18,49 students per year.
  • βœ… 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.
  • πŸ”’ To determine how fast the tuition is changing, find the derivative of the demand function with respect to time.
  • πŸ“‰ The rate of change of tuition (dp/dt) is negative, indicating that the tuition is decreasing.
  • πŸ’‘ The tuition is decreasing by up to $4.02 per credit hour per year.
  • πŸŽ“ 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
00:00
πŸ“š 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.

05:00
πŸ“‰ 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
πŸ’‘Related Rate Problem
A related rate problem is a type of mathematical problem where one must determine the rate at which a quantity is changing in relation to another changing quantity. In the video, it is used to find out how fast the tuition per credit hour is changing. The script states, '...this is a related rate problem, because we are looking for how fast, something is changing...'
πŸ’‘Tuition
Tuition refers to the fees that students pay to attend an educational institution. In the context of the video, tuition is the price per credit hour that students must pay, and it is the focus of the rate change analysis. The script mentions, '...what is the current tuition...'
πŸ’‘Demand Equation
The demand equation is a mathematical formula that relates the quantity demanded of a product to its price. In the video, the demand equation is 'q is equal to 440p, minus 0.7, p, squared', where 'q' is the number of students and 'p' is the price or tuition. It is central to solving the related rate problem.
πŸ’‘Rate of Change
The rate of change is a measure of how one quantity varies with respect to another, often expressed as a derivative in calculus. In the video, the rate of change is used to determine how fast the number of students (q) and the tuition (p) are changing over time. The script states, '...the rate of change of q, with respect to time, is 18, 49.'
πŸ’‘Quadratic Function
A quadratic function is a polynomial function of degree two, which can be represented in the standard form ax^2 + bx + c. In the video, the demand equation is a quadratic function, which is used to find the price of tuition. The script illustrates this by transforming the equation into a form that can be solved for 'p'.
πŸ’‘Derivative
In calculus, a derivative is a measure of how a function changes as its input changes. The derivative is used in the video to find the rate of change of the tuition price with respect to time. The script shows the process of taking the derivative of the demand function to solve for 'dp/dt'.
πŸ’‘Quadratic Formula
The quadratic formula is used to solve for the roots of a quadratic equation. In the video, it is mentioned as a method to find the value of 'p' in the demand equation, although the script opts to graph the equation instead. The quadratic formula is essential for solving related rate problems involving quadratic relationships.
πŸ’‘Graphing
Graphing is a visual method of plotting data points on a graph to understand the relationship between variables. The video script suggests graphing the demand equation to find the value of 'p' by identifying where the graph crosses the x-axis, which represents the solutions to the quadratic equation.
πŸ’‘Negative Price
A negative price in the context of the video would imply a situation where the educational institution is paying students to attend, which is not logical. The script dismisses the negative solution for 'p' as it does not make sense in the real-world scenario of tuition pricing.
πŸ’‘Decreasing Tuition
Decreasing tuition refers to a situation where the cost of attending an educational institution is going down over time. In the video, it is concluded that the tuition is decreasing by 'four dollars and two cents per credit hour per year', which is determined by the negative rate of change of tuition.
πŸ’‘Per Credit Hour
Per credit hour is a unit of measurement used to express the cost of educational courses, where the tuition is calculated based on the number of hours it takes to complete a credit. In the video, the focus is on how the tuition per credit hour is changing, which is central to the related rate problem being solved.
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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