Find the dimensions of a rectangle having a perimeter of 180 cm and an area that is as large as possible.
Let the length of the rectangle be \( l \) cm and the width be \( w \) cm.
Given that the perimeter of the rectangle is 180 cm:
\( 2l + 2w = 180 \)
\( l + w = 90 \)
The area of the rectangle is given by:
\( A = lw \)
Using the fact that the sum of the length and width is 90, we can express the area in terms of a single variable:
\( A = l(90 - l) = 90l - l^2 \)
To find the dimensions that maximize the area, we need to find the critical points of the function \( A \).
Differentiating \( A \) with respect to \( l \) and setting it equal to zero:
\( \frac{dA}{dl} = 90 - 2l = 0 \)
\( 2l = 90 \)
\( l = 45 \)
Therefore, the width of the rectangle is also 45 cm.
Therefore, the dimensions of the rectangle with the largest area would be Length = 45 cm and Width = 45 cm.