Question

How do you find the maximum area of a rectangle with perimeter of $40ft$?

Answer 1

Hint:Start by mentioning all the formulas that are necessary in these types of questions. Then next start by evaluating the perimeter of the rectangle then further evaluate the area of the rectangle. The perimeter and area of the rectangle are given by $2(l + b)$ and $A = l \times b$.

Complete step by step answer:
 First we will start off by mentioning the formula for the perimeter of the rectangle, which is given by $2(l + b)$ where $l$ is the length of the rectangle and $b$is the breadth or width of the rectangle. here, consider the values of length and breadth as $x,y$. Now, we will substitute the values of the terms in the above mentioned formula,
\[
P = 2(l + b) \\
\Rightarrow 40 = 2(x + y) \\
\Rightarrow 40 = 2x + 2y \\
\Rightarrow 20 = x + y \\
\Rightarrow y = 20 - x \\ \]
Now we will evaluate the area of the rectangle. Area of the rectangle is given by the formula,
$A = l \times b$ where $l$ is the length of the rectangle and $b$is the breadth or width of the rectangle. Now, we will substitute the values of the terms in the above mentioned formula,
$
A = x \times y \\
\Rightarrow A = x \times (20 - x) \\
\Rightarrow A = 20x - {x^2} \\ $
Now, here we have to evaluate an extreme for that.
We can do this by equating the value of the derivative equal to zero.
$
A = 20x - {x^2} \\
\Rightarrow A' = 20 - 2x \\ $
Now we will equate the derivative to zero.
$
20 - 2x = 0 \\
\Rightarrow 2x = 20 \\
\Rightarrow x = 10 \\ $
Here, we have evaluated the length of the rectangle and now we evaluate the width of the rectangle.
$
y = 20 - x \\
\Rightarrow y = 20 - 10 \\
\Rightarrow y = 10 \\ $
Hence, now we evaluate the area of the rectangle.
$
A = x \times y \\
\Rightarrow A = 10 \times 10 \\
\therefore A = 100 \\ $
Hence, the maximum area of the rectangle is $100\,\,sq.units$.

Note:While substituting the terms make sure you are taking into account the correct dimensions along with their units. Check if all the given terms have the same units, if not then convert all the terms to one single unit.The perimeter of a rectangle is the total length of all the sides of the rectangle. Hence, we can evaluate the perimeter by adding all four sides of a rectangle. Since opposite sides of a rectangle are always equal we need to evaluate only two sides to calculate the perimeter of the rectangle.