Problem 13 - Single Variable Calculus

Question

Satisfying Postal Requlations Postal regulations specify that a package sent by priority mail may have a combined length and girth of no more than 108 in. Find the dimensions of a cylindrical package with the greatest volume that may be sent by priority mail. What is the volume of such a package?

Step-by-Step Solution

Answer

The dimensions of the cylindrical package with the greatest volume that can be sent by priority mail are a height of 36 inches and a radius of \(\frac{108}{3π}\) inches. The maximum volume of such a package is \(1,728π\) cubic inches.

Step 1: Write the formula for the volume of a cylinder

The volume of a cylinder is given by the formula: \(V = π r^2 h\).

Step 2: Write the constraint for the length and girth

The combined length and girth of a package must be no more than 108 inches. For a cylinder, the length is the height, and the girth is the perimeter of the circular base, which is given by the formula \( 2πr\). Therefore, we have the equation: \( h + 2πr ≤ 108 \) or \( h = 108 - 2πr \).

Step 3: Substitute the constraint into the volume formula

Now, we'll substitute the expression for h from the constraint equation into the volume formula to eliminate h: \( V(r) = πr^2(108 - 2πr) \)

Step 4: Differentiate the volume function and find critical points

Now, we'll find the derivative of the volume function with respect to r: \( V'(r) = π[2r(108 - 2πr) - r^2(2π)] \) \( = 2π[108r - 2πr^2 - πr^2] \) \( = 2πr[108 - 3πr] \) Setting the derivative equal to zero, we find the critical points: \( 2πr[108 - 3πr] = 0 \) This yields two critical points: r = 0 and r = \(\frac{108}{3π}\)

Step 5: Determine which critical point maximizes the volume

Since r = 0 corresponds to a cylinder with no volume, we can discard this case. Thus, the critical point that maximizes the volume is r = \(\frac{108}{3π}\). Now, we can find the corresponding height h by substituting this value of r back into the h equation: \( h = 108 - 2π\frac{108}{3π} \) \( h = 108 - 72 = 36 \)

Step 6: Find the maximum volume

Now that we have the dimensions of the cylindrical package that maximize the volume, we can find the maximum volume by substituting the values of r and h into the volume formula: \( V_{max} = π(\frac{108}{3π})^2(36) \) \( V_{max} = 4π(108)^2/9 = 1,728π \) The dimensions of the cylindrical package with the greatest volume that can be sent by priority mail are a height of 36 inches and a radius of 108/(3π) inches. The maximum volume of such a package is 1,728π cubic inches.

Key Concepts

Cylinder Volume Formula

In order to solve problems involving cylinders, understanding the volume formula is crucial. Cylinders are three-dimensional shapes with two parallel circular bases connected by a curved surface. The formula for the volume of a cylinder is expressed as \( V = \pi r^2 h \). Here, \( r \) is the radius of the base, and \( h \) is the height of the cylinder. Before moving forward, make sure you remember these key elements to calculate the volume:

• \( \pi \) is a mathematical constant approximately equal to 3.14159.
• \( r \), which represents the radius, is the distance from the center of the circular base to its edge.
• \( h \) is the vertical distance from one base to the other, also known as the height.

Using this formula helps us understand how the size and shape of a cylinder influence its volume. This is an important starting point for problems involving optimization, as it gives a solid foundation to work with different dimensions.

Optimization in Calculus

Optimization is a powerful approach in calculus used to find the maximum or minimum value of a function under given conditions. For this particular problem, the aim is to maximize the volume of a cylindrical package while respecting certain postal constraints.
To handle this, we express the cylinder's volume in terms of both its radius and height. Then, by substituting the postal constraint for height (\( h = 108 - 2\pi r \)), we reduce the problem to a single variable. This allows us to write the volume solely as a function of the radius \( V(r) = \pi r^2(108 - 2\pi r) \).

Next, we apply differentiation to this volume function, finding the derivative \( V'(r) \), and setting it to zero to locate potential maximum or minimum points, known as critical points. Solving this provides the specific dimension (radius) that maximizes the volume under the given constraint. The calculus involved reveals that among the critical points, the radius \( \frac{108}{3\pi} \) provides the optimal solution for maximum volume, as it satisfies both volume and constraint requirements.

Postal Regulation Constraints for Packages

Postal regulations impose specific constraints to ensure packages can be handled safely and efficiently. In this case, the rule states that a combined length and girth (the perimeter of the base) of a package should not exceed 108 inches.

• The **length** of a cylindrical package is simply its height \( h \).
• The **girth** of the cylinder corresponds to the perimeter of the circular base, quantified as \( 2\pi r \) for a circle with radius \( r \).

To comply with these postal constraints, any dimensions chosen for our cylinder must ensure that the sum of the height and girth does not exceed 108 inches. As part of the optimization process, these constraints are translated into mathematical equations that limit and guide us towards acceptable dimensions, specifically: \[ h + 2\pi r \leq 108 \].
By considering such regulations, we can identify feasible dimensions for mailing packages that meet all required standards while maximizing functionality, such as maximizing the volume in this scenario. It’s essential to integrate these constraints early in the problem-solving process to prevent choosing dimensions that are not permissible for mailing.