# Collin's question at Yahoo Answers regarding maximization of beam strength

• MHB
• MarkFL
In summary, we have solved the Calculus Max Min Application Problem by finding the dimensions of the strongest rectangular beam that could be cut from a log whose cross section is a circle of the form x^2+y^2=256. The optimal dimensions are D=32√(2/3) and W=32/√3, and the objective function is S(D,W)=kD^2W.
MarkFL
Gold Member
MHB
Calculus Max Min Application Problem?

If the strength of a rectangular beam is proportional to the product of its width and the square of its depth, find the dimensions of the strongest beam that could be cut from a log whose cross section is a circle of the form x^2+y^2=256

I have posted a link there to this thread so the OP can view my work.

Hello Collin,

Let's let $0<W$ be the width (horizontal dimension) of the beam's cross-section and $0<D$ be the depth (vertical dimension). The strength $S$ of the beam is our objective function:

$$\displaystyle S(D,W)=kD^2W$$ where $$\displaystyle 0<k\in\mathbb{R}$$ is the constant of proportionality.

The vertices of the beam's cross-section must obviously confined to the circle:

$$\displaystyle x^2+y^2=16^2$$

Hence:

$$\displaystyle W=2x\implies x=\frac{W}{2}$$

$$\displaystyle D=2y\implies y=\frac{D}{2}$$

And so we may write:

$$\displaystyle \left(\frac{W}{2} \right)^2+\left(\frac{D}{2} \right)^2-16^2=0$$

Thus, we may express the constraint as:

$$\displaystyle g(D,W)=D^2+W^2-32^2=0$$

Solving the constraint for $D^2$, we obtain:

$$\displaystyle D^2=32^2-W^2$$

Substituting into the objective function, we find:

$$\displaystyle S(W)=k\left(32^2-W^2 \right)W=k\left(32^2W-W^3 \right)$$

Differentiating with respect to $W$ and equating the result to zero, we find:

$$\displaystyle S'(W)=k\left(32^2-3W^2 \right)=0$$

Taking the positive root, we obtain the critical value:

$$\displaystyle W=\frac{32}{\sqrt{3}}$$

Using the second derivative test to determine the nature of the extremum associated with this critical value, we find:

$$\displaystyle S''(W)=-6kW$$

Since $$\displaystyle 0<W$$ then $$\displaystyle S''(W)<0$$ demonstrating that we have found a relative maximum of the objective function. Hence, the dimensions which maximize the beam's strength are:

$$\displaystyle D\left(\frac{32}{\sqrt{3}} \right)=\sqrt{32^2-\left(\frac{32}{\sqrt{3}} \right)^2}=32\sqrt{\frac{2}{3}}$$

$$\displaystyle W=\frac{32}{\sqrt{3}}$$

We could also use a multi-variable technique, Lagrange multipliers.

We have the objective function:

$$\displaystyle S(D,W)=kD^2W$$

subject to the constraint:

$$\displaystyle g(D,W)=D^2+W^2-32^2=0$$

giving rise to the system:

$$\displaystyle 2kDW=\lambda(2D)$$

$$\displaystyle kD^2=\lambda(2W)$$

Thus:

$$\displaystyle \lambda=kW=\frac{kD^2}{2W}\implies D^2=2W^2$$

substituting into the constraint, we obtain:

$$\displaystyle 2W^2+W^2-32^2=0$$

$$\displaystyle W=\frac{32}{\sqrt{3}}$$

$$\displaystyle D=\sqrt{2}W=32\sqrt{\frac{2}{3}}$$

## 1. What is beam strength and why is it important?

Beam strength refers to the ability of a beam or structure to resist bending or breaking under a load. It is important in engineering and construction as it ensures the safety and stability of a structure.

## 2. How can beam strength be maximized?

Beam strength can be maximized through various methods such as increasing the beam's dimensions, using stronger materials, and designing the structure to distribute the load evenly. Other factors such as proper support and reinforcement also play a role in maximizing beam strength.

## 3. What are the factors that affect beam strength?

The factors that affect beam strength include the beam's dimensions, material properties, support conditions, and the type and direction of the load. Other factors such as temperature and environmental conditions may also have an impact.

## 4. Can beam strength be increased indefinitely?

No, beam strength cannot be increased indefinitely. There are limitations based on the properties of the materials being used and the structural design. However, it can be optimized to meet specific needs and requirements.

## 5. Why is it important to consider beam strength in structural design?

Considering beam strength in structural design is crucial as it ensures the safety and stability of the structure. A structure with inadequate beam strength may fail under load, leading to potential disasters and safety hazards. It also affects the longevity and durability of the structure.

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