Question

among all rectangles that have a perimeter of 60, find the dimensions of the one whose area is largest. write your answers as fractions reduced to lowest terms.

Explanation:

Step1: Define variables

Let the length of the rectangle be $l$ and the width be $w$. The perimeter formula is $P = 2(l + w)$. Given $P=60$, so $2(l + w)=60$, which simplifies to $l + w=30$, and we can express $l = 30 - w$.

Step2: Set up area function

The area formula of a rectangle is $A=l\times w$. Substitute $l = 30 - w$ into the area formula, we get $A(w)=(30 - w)w=30w - w^{2}$.

Step3: Find the maximum of the function

This is a quadratic - function $y = ax^{2}+bx + c$ with $a=-1$, $b = 30$, $c = 0$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its $x$ - coordinate (in our case $w$ - coordinate) at $w=-\frac{b}{2a}$. Substitute $a=-1$ and $b = 30$ into the formula, we have $w=-\frac{30}{2\times(-1)} = 15$.

Step4: Find the length

Since $l + w=30$ and $w = 15$, then $l=30 - 15=15$.

Answer:

Length: $15$, Width: $15$