Question

Picnic Pavilion Rental The members of the Blue Boar society always divide the pavilion ree for their picnics equally among the members. Currently there are 65 members and the pavilion rents for \(\$ 250 .\) The pavilion cost is increasing at a rate of \(\$ 10\) per year, while the Blue Boar membership is increasing at a rate of 6 members per year. What is the current (instantaneous) rate of change in each member's share of the pavilion rental fee? Answer in appropriate units of measure.

Short answer

The instantaneous rate of change in each member's share of the pavilion rental fee is approximately -\$0.008 per member per year.

Step-by-step solution

Expression for members' share

Calculate each member's share of the pavilion rental. This can be done by dividing the cost of the pavilion \(P\) by the number of members \(M\). So, each member's share \(S\) can be expressed as \(S = P/M\).

Differentiation

Differentiate both sides with respect to time \(t\) to obtain an expression for the rate of change of \(S\) with respect to \(t\). This is done using the quotient rule for differentiation, which gives \(\frac{dS}{dt} = \frac{M \cdot \frac{dP}{dt} - P \cdot \frac{dM}{dt}}{{M^2}}\).

Substitute

Substitute the given values of \(P = 250\), \(M = 65\), \(\frac{dP}{dt} = 10\), and \(\frac{dM}{dt} = 6\) into \(\frac{dS}{dt} = \frac{M \cdot \frac{dP}{dt} - P \cdot \frac{dM}{dt}}{{M^2}}\) to find the value of \(\frac{dS}{dt}\).

Calculate

Upon substituting the values into the equation, it can be simplified to obtain the instantaneous rate of change in rental fee per member, which will give the answer in terms of dollars per member per year.

Key concepts

Quotient Rule for Differentiation

The quotient rule is a technique used in calculus to find the derivative of a function that is the quotient of two differentiable functions. Consider the function represented as \( \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are both differentiable. The quotient rule states that the derivative of this function with respect to \( x \) is given by:
\[ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{\left(g(x)\right)^2} \]
Applying this to real-world problems, such as determining how the cost per person changes when both the cost and the number of people change over time, is a common application. By following the steps of differentiation and substituting the respective rate of changes (\(\frac{dP}{dt}\) for the pavilion cost \( P \) and \(\frac{dM}{dt}\) for the number of members \( M \) ), the quotient rule enables us to find how the share per member is affected instantaneously.

Differentiation

Differentiation is a fundamental operation in calculus that deals with how a function changes as its input changes. It's the process of finding the derivative of a function, which represents the rate at which the function's value changes at any given point. In practical terms, differentiation can inform us about rates of change in various phenomena, such as speed, growth, or decay.
For our example of the Blue Boar society, differentiation allows us to understand how each member's share of the pavilion rental fee changes as the number of members and the rental fee itself change over time.

Calculus Applications

Calculus has numerous applications in science, engineering, economics, and beyond. It helps model and solve problems involving dynamic changes. In our exercise concerning the rental fee for a picnic pavilion, calculus is used to ascertain the members' cost-sharing dynamics. The instantaneous rate of change—a concept calculus excels at—is particularly useful for predicting future trends based on current rates of change. Thus, calculus not only helps solve problems but also provides insights into the relationships between changing quantities.

Related Rates Problems

Related rates problems are a type of problem in calculus where two or more related quantities are changing with respect to time, and you are interested in how the rates of change affect each other. These problems typically involve the application of the chain rule of differentiation and often require the quotient rule as well when dealing with division of functions.
For the Blue Boar society, the related rates problem involves understanding how the changing number of members and increasing rental fee will affect each member's share of the cost. By establishing the relationship between these rates of change using differentiation, we can solve for the instantaneous rate of change that each member will experience in their share of the rental fee.