Question

A farmer has 100 m of fencing to enclose a rectangular pen. Which quadratic equation gives the area (A) of the pen, given its width (w)?

Answer 1

Answer:

$A(w)=50w-w^2$

Step-by-step explanation:

Let w be the width of rectangular pen.

We have been given that a framer has 100 m of fencing to enclose a rectangular pen. This means that perimeter of pen is 100 meter.

Since the perimeter is 2 times the length and width of rectangle.

$\text{Perimeter of rectangle}=2\text{ (Width+Length})$

Upon substituting our given values in above formula we will get,

$100=2(w\text{+Length})$

$\frac{100}{2}=\frac{2(w\text{+Length})}{2}$

$50=w\text{+Length}$

$50-w=w-w\text{+Length}$

$50-w=\text{Length}$

$\text{Area of rectangle}=\text{Width*Length}$

Upon substituting our given values in area formula we will get,

$\text{Area of rectangle}=w(50-w)$

$\text{Area of rectangle}=50w-w^2$

Let us represent area in terms of width of rectangle as:

$A(w)=50w-w^2$

Therefore, the area of our given pen will be $A(w)=50w-w^2$.

Answer 2

A(w)=50w-w2 if the perimeter is 100, I plugged in possible values for the width...I used w=15 and length = 35 for a perimeter of 100. then I plugged these values in the possible equations until I found the correct one giving me an area of 525 (15x35) . I could not think of an easier way.