Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Find the point of the sphere \(x^{2}+y^{2}+z^{2}=25\) that is closest to (3,4,5) .

Step-by-Step Solution

Answer

The point closest to (3,4,5) is \(\left( \frac{3}{11}, \frac{4}{11}, \frac{5}{11} \right)\).

Step 1: Understanding the Problem

We need to find the point on the sphere given by the equation \(x^2 + y^2 + z^2 = 25\) that is closest to the point \((3, 4, 5)\).

Step 2: Setting Up the Objective Function

Our objective is to minimize the distance from a point \((x, y, z)\) on the sphere to the point \((3, 4, 5)\). The distance squared is given by \[D^2 = (x - 3)^2 + (y - 4)^2 + (z - 5)^2.\]

Step 3: Using Lagrange Multipliers

We use the method of Lagrange multipliers to minimize \(D^2\) subject to the constraint \(x^2 + y^2 + z^2 = 25\). Define \[ f(x, y, z) = (x - 3)^2 + (y - 4)^2 + (z - 5)^2 \] and \[ g(x, y, z) = x^2 + y^2 + z^2 - 25. \] Set the gradients equal: \[ abla f = \lambda abla g. \]

Step 4: Calculating Gradients

Calculate gradients: \[ abla f = (2(x - 3), 2(y - 4), 2(z - 5)) \] and \[ abla g = (2x, 2y, 2z). \]

Step 5: Setting Up Equations

From \(abla f = \lambda abla g\), the equations are:1. \(2(x - 3) = \lambda 2x\)2. \(2(y - 4) = \lambda 2y\)3. \(2(z - 5) = \lambda 2z\)

Step 6: Solving System of Equations

Solve for \(\lambda\):1. \(x - 3 = \lambda x\)2. \(y - 4 = \lambda y\)3. \(z - 5 = \lambda z\)Rearranging gives:\(x(1 - \lambda) = 3, y(1 - \lambda) = 4, z(1 - \lambda) = 5\).

Step 7: Finding \(\lambda\) and Plugging Back

Using \(x^2 + y^2 + z^2 = 25\), substitute \(x = \frac{3}{1 - \lambda}, y = \frac{4}{1 - \lambda}, z = \frac{5}{1 - \lambda}\).Plug these into the sphere equation and solve for \(\lambda\).

Step 8: Substitute and Calculate Coordinates

After solving for \(\lambda\), substitute back to get the point:\[(x, y, z) = \left( \frac{3}{11}, \frac{4}{11}, \frac{5}{11} \right).\]

Key Concepts

Lagrange multipliers

When you need to find the extreme values (like a minimum or maximum) of a function subject to a constraint, the method of Lagrange multipliers is a powerful tool.
This technique involves introducing a new variable, known as the multiplier, to deal with the constraint.In this exercise, we are given the constraint of the sphere equation, and we are tasked with minimizing the distance to a point.
Here's how the Lagrange multiplier method works:

• Define the function you want to optimize, called the objective function. In our case, it's the distance squared from a point on the sphere to the fixed point.
• Define the constraint as another function. For the sphere, this is the equation of the sphere itself.
• Calculate the gradients of both the objective function and the constraint function.
• Set these gradients equal to each other, multiplied by a constant known as the Lagrange multiplier (denoted usually by \(\lambda\)).
  This gives you a system of equations that includes both the functions and the constraint.

The magic of this method is in how it incorporates the constraint directly into the optimization process, allowing us to solve for the minimal (or maximal) points directly along the constraint line.

Distance minimization

Minimizing distance is a common problem in mathematics, physics, and engineering, where often you wish to find the shortest path or closest approach between points.
In mathematical terms, the shortest distance between two points is usually calculated using a direct formula that stems from the Pythagorean theorem. In our exercise, we're looking at minimizing the square of the distance between a variable point (currently unspecified) on a sphere and a specific fixed point (3, 4, 5) in space.

• The distance squared formula is used instead of the direct distance to simplify calculations, as it avoids dealing with square roots.
• The function to minimize is the square of the Euclidean distance:\[D^2 = (x - 3)^2 + (y - 4)^2 + (z - 5)^2\]. This expression expands the traditional distance formula by squaring the components, which highlights the separation in each dimension.

By minimizing this squared distance on the sphere, we've ensured that the point found is the closest one to (3, 4, 5) while still satisfying the sphere's geometric constraints.

Sphere equation constraint

The equation of a sphere is a typical form used to represent circular objects in three-dimensional space, generally given as \( x^2 + y^2 + z^2 = r^2 \), where \( r \) is the radius.
This constraint ensures that any solution derived will stay on the surface of the sphere, maintaining its defined size and location in space.In our specific case, the sphere's equation is \( x^2 + y^2 + z^2 = 25 \), indicating that the sphere:

• Is centered at the origin (0, 0, 0) in the coordinate system.
• Has a radius of 5, since \( r^2 = 25 \) implies that the radius \( r \) is 5.

The condition offered by the equation, therefore, acts as a boundary that our solution — the closest point — must satisfy.
By solving the system of equations presented by the Lagrange method, we're able to ensure that our optimized point lies exactly on this spherical surface, meeting the criteria for both proximity to the target point and position within the sphere's defined area.