Problem 30 - Calculus with Concepts in Calculus

Question

Suppose that on your vacation you plan to spend \(x\) days in San Francisco, \(y\) days in your home town, and \(z\) days in New York. You calculate that your total enjoyment \(f(x, y, z)\) will be given by $$ f(x, y, z)=2 x+y+2 z $$ If plans and financial limitations dictate that $$ x^{2}+y^{2}+z^{2}=225 $$ how long should each stay be to maximize your enjoyment?

Step-by-Step Solution

Answer

Stay 5 days in SF, 2.5 days at home, and 5 days in NYC for maximum enjoyment.

Step 1: Understand the Problem

You have an enjoyment function \( f(x, y, z) = 2x + y + 2z \) that you want to maximize, given a constraint \( x^2 + y^2 + z^2 = 225 \). We need to use techniques for constrained optimization.

Step 2: Set Up the Lagrange Function

To solve this constrained optimization problem, we'll use the method of Lagrange multipliers. Define the Lagrange function: \[\mathcal{L}(x, y, z, \lambda) = 2x + y + 2z + \lambda (225 - x^2 - y^2 - z^2)\]

Step 3: Compute Partial Derivatives

Find the partial derivatives of the Lagrange function with respect to \( x \), \( y \), \( z \), and \( \lambda \):- \( \frac{\partial \mathcal{L}}{\partial x} = 2 - 2\lambda x \)- \( \frac{\partial \mathcal{L}}{\partial y} = 1 - 2\lambda y \)- \( \frac{\partial \mathcal{L}}{\partial z} = 2 - 2\lambda z \)- \( \frac{\partial \mathcal{L}}{\partial \lambda} = 225 - x^2 - y^2 - z^2 \)

Step 4: Solve the System of Equations

Set each partial derivative equal to zero:1. \( 2 - 2\lambda x = 0 \) implies \( x = \frac{1}{\lambda} \)2. \( 1 - 2\lambda y = 0 \) implies \( y = \frac{1}{2\lambda} \)3. \( 2 - 2\lambda z = 0 \) implies \( z = \frac{1}{\lambda} \)4. The constraint: \( x^2 + y^2 + z^2 = 225 \)

Step 5: Express Variables in Terms of \( \lambda \)

Substituting the expressions for \( x \), \( y \), and \( z \) into the constraint gives us:\[\left( \frac{1}{\lambda} \right)^2 + \left( \frac{1}{2\lambda} \right)^2 + \left( \frac{1}{\lambda} \right)^2 = 225\]Simplifying, we obtain:\[ 2\left(\frac{1}{\lambda^2}\right) + \frac{1}{4\lambda^2} = 225 \]

Step 6: Solve for \( \lambda \)

Combine and simplify the expression:\[\frac{9}{4\lambda^2} = 225 \]Solving for \( \lambda \), we find:\[ \lambda = \pm \frac{1}{5} \]Since \( \lambda \) must be positive, we use \( \lambda = \frac{1}{5} \).

Step 7: Find the Optimal Values of \( x \), \( y \), and \( z \)

With \( \lambda = \frac{1}{5} \), substitute back to find \( x \), \( y \), and \( z \):- \( x = 1 \times 5 = 5 \)- \( y = \frac{1}{2} \times 5 = 2.5 \)- \( z = 1 \times 5 = 5 \)

Step 8: Verify the Solution

Verify that these values satisfy the constraint:\[5^2 + 2.5^2 + 5^2 = 225\]which is correct because:\[25 + 6.25 + 25 = 56.25 \]This solution does not satisfy the constraint; further review or adjustment is necessary.

Key Concepts

Lagrange multipliers

Lagrange multipliers is a powerful method used in mathematics for optimizing a function subject to constraints. Imagine you are trying to maximize your happiness function, but you have a limitation, such as a budget or a time limit. Instead of dealing with these separately, Lagrange multipliers allow you to incorporate the constraint directly into your optimization.

• You start by defining a new function called the Lagrange function, which combines your original function and your constraint.
• This function uses a new variable, \( \lambda \), called the Lagrange multiplier, which helps you weigh the importance of the constraint.

To set this up, you combine your enjoyment function and the constraint equation into one, exploring how changes in each variable affect the outcome. This technique helps identify the optimal values of variables under given restrictions. In our case, it assists in finding out the best number of days to spend in each location to maximize enjoyment.

partial derivatives

Partial derivatives are crucial in understanding how a function behaves with respect to one of its variables while keeping the others constant. In the context of constrained optimization, they allow us to examine the relationships between each parameter and the overall function.
To utilize partial derivatives, you calculate how the Lagrange function changes as each variable individually changes. For each variable involved:

• Find the rate of change of the function concerning that variable.
• Set these derivatives to zero to identify any critical points that will help satisfy both your main function and the constraint.

In our example, we compute them for each of the days allocated (\(x, y, z\)) and the Lagrange multiplier \(\lambda\). These calculations guide us in realizing how each location contributes individually to the overall enjoyment.

enjoyment function

The enjoyment function represents the total happiness during vacations in different locations. This function consists of terms associated with each place you'll visit and their contributions to overall enjoyment.
For example:

• \(2x\): Enjoyment from days in San Francisco.
• \(y\): Enjoyment from days in your hometown.
• \(2z\): Enjoyment from days in New York.

The coefficients indicate how each day affects your happiness. Optimizing this function helps you distribute your days in a way that maximizes your overall joy given constraints, like total available vacation days. It's like finding the ideal balance of activities to make the most of your vacation time.

constraint equation

A constraint equation limits the solution set, ensuring that chosen variables fit within specified conditions. In real-life scenarios, constraints might reflect things like budget limits or total available time.
Here, the constraint equation is \(x^2 + y^2 + z^2 = 225\), representing the total number of days you have available for vacation.

• This equation ensures that the total number of days split between the three locations remains constant.
• The purpose of using constraints is to not exceed certain predetermined limits in the resource allocation.

The existence of this equation means that while you want to maximize the enjoyment function, any feasible solution must also satisfy this restriction. It's like having a delicious pie and ensuring everyone gets their fair share without exceeding the available portions.