Question

A fibre of length \( l \) is to be divided into two pieces. Multiplying the square of the length of one piece with the cube of the length of the other piece yields the greatest possible product value. Amongst the following, the correct combination of the lengths is:

• A.
• B.
• C.
• D.

Hint:

When maximizing or minimizing products of lengths, use differentiation to find critical points and then check the value of the function.

Answer 1

The Correct Option is A

Step 1: Define the lengths of the two pieces. 
Let the lengths of the two pieces be \( x \) and \( l - x \). 
Step 2: Define the product function. 
The product \( P \) of the square of one piece and the cube of the other piece is given by: \[ P(x) = x^2 (l - x)^3 \] Step 3: Maximize the product. 
To find the value of \( x \) that maximizes \( P(x) \), we differentiate \( P(x) \) with respect to \( x \): \[ \frac{dP}{dx} = 2x(l - x)^3 - 3x^2(l - x)^2 \] Setting \( \frac{dP}{dx} = 0 \) to find critical points, we solve for \( x \). After solving, we find that the value of \( x = \frac{2l}{5} \) and \( l - x = \frac{3l}{5} \) yields the greatest product. Step 4: Conclusion. 
Thus, the correct combination of the lengths is \( \boxed{\frac{2l}{5} { and } \frac{3l}{5}} \).