you need to construct a fence around a an area of 1936 m² to build a garden. what are the dimensions x and y…

Explanation:

Step1: Set up area and perimeter equations

Let the area of the rectangle be $A = xy=1936$, so $y=\frac{1936}{x}$. The perimeter $P = 2x + 2y$. Substitute $y=\frac{1936}{x}$ into the perimeter formula, we get $P(x)=2x + 2\times\frac{1936}{x}=2x+\frac{3872}{x}$.

Step2: Find the derivative

Differentiate $P(x)$ with respect to $x$. Using the power - rule, if $y = ax^n$, then $y^\prime=anx^{n - 1}$. So $P^\prime(x)=2-\frac{3872}{x^{2}}$.

Step3: Set the derivative equal to zero

Set $P^\prime(x)=0$ to find the critical points. [ \begin{align*} 2-\frac{3872}{x^{2}}&=0\ 2&=\frac{3872}{x^{2}}\ 2x^{2}&=3872\ x^{2}& = 1936\ x&=44 \end{align*} ] We take the positive value since $x$ represents a length.

Step4: Find the value of y

Substitute $x = 44$ into $y=\frac{1936}{x}$. Then $y=\frac{1936}{44}=44$.

Answer:

$x = 44$ m $y = 44$ m