The stiffness of a beam of rectangular cross-section varies as the product of the breadth and square of the depth. What must be the breadth and depth of the stiffest beam that can be cut from a leg of diameter 'd'?

Answer

To solve the problem, we need to find the breadth (B) and depth (H) of the stiffest beam that can be cut from a circular log of diameter D. The stiffness of the beam is given as proportional to the product of the breadth and the square of the depth, which can be expressed mathematically as:

\[ I = K \cdot B \cdot H^2 \]

where K is a constant.

### Step 1: Relate the dimensions of the beam to the diameter of the log
Since the beam is cut from a circular log, we can use the Pythagorean theorem to relate the breadth and depth to the diameter of the log. The relationship is given by:

\[ D^2 = B^2 + H^2 \]

### Step 2: Express H in terms of B
From the equation above, we can express H in terms of B:

\[ H^2 = D^2 - B^2 \]
\[ H = \sqrt{D^2 - B^2} \]

### Step 3: Substitute H into the stiffness equation
Now we substitute H into the stiffness equation:

\[ I = K \cdot B \cdot H^2 \]
\[ I = K \cdot B \cdot (D^2 - B^2) \]

### Step 4: Differentiate I with respect to B
To find the value of B that maximizes the stiffness I, we differentiate I with respect to B and set the derivative equal to zero:

\[ \frac{dI}{dB} = K \cdot (D^2 - B^2) + K \cdot B \cdot (-2B) = K \cdot (D^2 - 3B^2) \]

Setting the derivative equal to zero:

\[ K \cdot (D^2 - 3B^2) = 0 \]

### Step 5: Solve for B
Since K is a constant and cannot be zero, we can simplify the equation:

\[ D^2 - 3B^2 = 0 \]
\[ D^2 = 3B^2 \]
\[ B^2 = \frac{D^2}{3} \]
\[ B = \frac{D}{\sqrt{3}} \]

### Step 6: Find H using the value of B
Now we can find H using the value of B we just calculated:

\[ H^2 = D^2 - B^2 \]
\[ H^2 = D^2 - \frac{D^2}{3} \]
\[ H^2 = \frac{3D^2}{3} - \frac{D^2}{3} \]
\[ H^2 = \frac{2D^2}{3} \]
\[ H = \sqrt{\frac{2D^2}{3}} = \frac{D\sqrt{2}}{\sqrt{3}} \]

### Final Answer
Thus, the dimensions of the stiffest beam that can be cut from a leg of diameter D are:

- Breadth \( B = \frac{D}{\sqrt{3}} \)
- Depth \( H = \frac{D\sqrt{2}}{\sqrt{3}} \)