Question

use the previous question and applet to answer the following questions. drag the purple x to vary the value of the length of the rectangular fence. let ( a = g(l)).
c. use the above graph to estimate the dimensions of the rose garden that has a perimeter of 300 feet and an area of 5,200 square feet. let ( l < w).
( l = 54.4) feet preview
( w = 95.6) feet preview
d. complete the following statement.
the rectangular rose garden (using 300 feet of fence) has a maximum area of square feet when the rectangle has the following dimensions:
( l =) feet preview
( w =) feet preview

Explanation:

Step1: Recall perimeter formula

For a rectangle, perimeter $P = 2(l + w)$. Given $P=300$, so $l + w=150$, then $w = 150 - l$.

Step2: Recall area formula

Area $A=l\times w=l(150 - l)=150l - l^{2}$. This is a quadratic - function of the form $y = ax^{2}+bx + c$ where $a=-1$, $b = 150$ and $c = 0$.

Step3: Find the vertex of the quadratic function

The $x$ - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. For $A=-l^{2}+150l$, $l=-\frac{150}{2\times(-1)} = 75$.

Step4: Find the maximum area

Substitute $l = 75$ into the area formula $A=-l^{2}+150l$. So $A=-(75)^{2}+150\times75=-5625 + 11250=5625$ square feet. And $w=150 - 75 = 75$ feet.

Answer:

The rectangular rose - garden (using 300 feet of fence) has a maximum area of 5625 square feet when the rectangle has the following dimensions:
$l = 75$ feet
$w = 75$ feet