Question

Assume that the connecting rod and the crank of an engine forms as two sides of the triangle. If the area included in the triangle is maximum when the crank is at $60^\circ$, then the ratio of length of connecting rod to the radius of the crank is

• A. $1$
• B. $1.414$
• C. $1.732$
• D. $0.577$

Hint:

When the area is maximized, the crank and connecting rod form a right angle ($90^\circ$). This creates a standard $30^\circ-60^\circ-90^\circ$ right triangle, where the ratio of the opposite side to the adjacent side is always $\sqrt{3} \approx 1.732$.

Answer 1

The Correct Option is C

Step 1: Look at the triangle in terms of base and height.
Take the crank \( OB = r \) as one side, with the far end of the connecting rod \( C \) tracing out the area as the crank turns. For a fixed side \( OB \) and a fixed rod length \( BC = l \), the triangle's area is largest when \( C \) sits as far as possible from the line through \( OB \), which happens exactly when \( BC \) is perpendicular to \( OB \).

Step 2: Recognise the right triangle this creates.
So at maximum area, triangle \( OBC \) has a right angle at \( B \), while the crank angle \( \angle BOC = 60^\circ \) sits at vertex \( O \). In this right triangle, the side opposite \( 60^\circ \) is \( BC = l \) and the side adjacent to it is \( OB = r \).

Step 3: Apply the tangent ratio.
\[ \frac{l}{r} = \tan(60^\circ) = \sqrt{3} \approx 1.732 \] \[ \boxed{1.732} \] This matches option (C).