Problem 34 - Thomas Calculus

Question

Strength of a beam The strength \(S\) of a rectangular wooden beam is proportional to its width times the square of its depth. (See accompanying figure.) a. Find the dimensions of the strongest beam that can be cut from a 12 -in.- diameter cylindrical log. b. Graph \(S\) as a function of the beam's width \(w\) , assuming the proportionality constant to be \(k=1 .\) Reconcile what you see with your answer in part (a). c. On the same screen, graph \(S\) as a function of the beam's depth \(d,\) again taking \(k=1 .\) Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of \(k ?\) Try it.

Step-by-Step Solution

Answer

Strongest beam: \( w = \sqrt{48}, d = \sqrt{96} \). Adjusting \( k \) scales strength.

Step 1: Understand the Problem

We need to find the dimensions of the strongest beam that can be cut from a cylindrical log. The strength \( S \) is proportional to \( w \times d^2 \), where \( w \) is the width and \( d \) is the depth. The log has a diameter of 12 inches.

Step 2: Set Up the Equation for the Beam's Cross-Section

The beam's cross-section must fit inside a circle of diameter 12 inches. Thus, its equation must satisfy the condition \( \frac{w}{2}^2 + \frac{d}{2}^2 = 6^2 \). This can be written as \( w^2 + d^2 = 144 \).

Step 3: Express \( S \) as a Function of One Variable

To find a single function for \( S \), express \( d \) in terms of \( w \) using the equation of the circle: \( d = \sqrt{144 - w^2} \). Substitute this into \( S = k \cdot w \cdot d^2 \) to get \( S(w) = k \cdot w \cdot (144 - w^2) \).

Step 4: Differentiate to Find Maximum Strength

Set \( k = 1 \) for simplicity, and write \( S(w) = w(144 - w^2) \). Differentiate \( S \) with respect to \( w \): \( \frac{dS}{dw} = 144 - 3w^2 \). Set \( \frac{dS}{dw} = 0 \) and solve for \( w \). We find \( w = \sqrt{48} \).

Step 5: Solve for Depth \( d \)

Using \( w = \sqrt{48} \), substitute back into the circle's equation: \( d^2 = 144 - 48 = 96 \), hence \( d = \sqrt{96} \).

Step 6: Graph \( S \) as a Function of \( w \) and \( d \)

Graph \( S(w) = w(144 - w^2) \). Observe that this function peaks at \( w = \sqrt{48} \). Similarly, graph \( S(d) \) using \( w = \sqrt{144 - d^2} \) and find it also peaks at \( d = \sqrt{96} \).

Step 7: Analyze Effect of Changing \( k \)

Changing \( k \) scales the entire function \( S(w) \) or \( S(d) \) proportionally. It changes the scale of the strength but does not change the dimensions of the strongest beam.

Key Concepts

Beam strength calculation

When calculating the strength of a beam, particularly a rectangular wooden one, a unique relationship exists. The strength, denoted by \( S \), is directly proportional to the product of the beam's width \( w \) and the square of its depth \( d \). This can be expressed mathematically as \( S = k \cdot w \cdot d^2 \), where \( k \) is a proportionality constant. This formula highlights how both the width and depth of the beam influence its strength. An increase in either dimension leads to an increase in the beam's strength. However, because the depth is squared, changes in the depth have a larger effect on the beam's strength than the same changes in width do. Understanding this formula is crucial in determining how to optimize the dimensions of a beam to maximize its strength.

Rectangular beam

A rectangular beam is a common structural element characterized by having a rectangular cross-section. In the context of our problem, we consider a beam cut from a 12-inch diameter cylindrical log. This constraint means the beam's cross-section must fit inside the circle with a diameter like the log. The dimensions of this rectangle, width \( w \) and depth \( d \), are crucial since they influence the beam's strength. The cross-section has to satisfy the equation for a circle because any beam with a larger cross-section would not fit within the log. So, the constraint becomes \( w^2 + d^2 = 144 \). This equation governs the possible widths and depths the beam can have that will still fit within the log's boundaries.

Proportionality constant

The proportionality constant \( k \) in the beam strength formula plays a role in determining the actual numerical value of the beam's strength. However, it does not affect the relationship between the beam's width and depth in this optimization problem. By setting \( k = 1 \), we simplify the calculations without altering the optimal dimensions that maximize strength. Changing \( k \) would uniformly scale the calculated strength, increasing or decreasing all calculations by the factor \( k \). But it is important to note that varying \( k \) would not change where the strength maximizes on a graph of \( S(w) \) or \( S(d) \). Hence, for practical purposes of determining dimensions, \( k \) serves more as a scaling reference.

Graphical analysis in calculus

Graphical analysis in calculus provides insight into how changing variables affect outcomes like beam strength. By graphing the function \( S(w) = w(144 - w^2) \), we can visually determine where the function peaks, helping identify the optimal dimensions that maximize strength. Differentiation allows us to find the peak by locating where the derivative is zero, indicating a maximum point. Calculus helps in computing this by setting \( \frac{dS}{dw} = 0 \), resulting in \( w = \sqrt{48} \).
Visual analysis of the graph shows how \( S \) behaves as \( w \) or \( d \) varies, and corroborates the calculated maximum from the differentiation. By also graphing \( S(d) \) with \( d = \sqrt{96} \), this analysis confirms consistency across different forms of the function. Such graphical insights are vital for validating analytical findings and understanding how dimensions affect beam performance.