Question

Jessica has 900 feet of fencing and a very large field. She can enclose a rectangular area with dimensions x feet and (450 − x) feet. What is the largest rectangular area she can create

Answer 1

Answer:

The largest rectangular area is 225 feet by 225 feet giving an area of 50625 square feet

Explanation:

As the total fencing available is 900 feet, the area he available fencing can enclose will measure 'x' by '450-x' feet.

$L=x$

$W=450-x$

The area of a rectangle is determined by multiplying the length of perpendicular sides:

$A = L*W$

$A = x*(450-x)$

$A = 450x-x^{2}$

The derivative of an equation can be used to determine the slope at any given point of that equation. The slope will be zero at the maximum or minimum point of the equation. Thus, differentiating the equation for area and equating it to zero will give the value of 'x' where the area is maximum.

A simple variable can be differentiated using below concept:

$f(a)=a^{b}$

$f'(a)=ba^{b-1}$

Using the above concepts to differentiate equation for 'A' and equating to zero to calculate 'x' will give:

$A = 450x-x^{2}$

$A' = 450x^{1-1}-2x^{2-1}$

$A' = 450-2x=0$

$450-2x=0$

$x=225$

$L=x=225 feet$

$W=450-x=450-225=225 feet$

Answer 2

P = 2x+2(450-x)=900
A = x(450 - x)
A = x(450 - x)
-x² + 450x = A
-x² +450x = A
A is going to be maximum, when x is a vertex.
-(x² - 450x) = A
-(x² - 2*225x+225²) - 225² =A
-(x-225)² -225² = A
Vertex when x=225

Largest are 225(450-225) = 225² = 50625 ft²

The square has a largest area.