Question

part ii
answer the following:
the sum of the length ( l ) and the width ( w ) of a rectangular area is 240 feet.

a. write ( w ) as a function of ( l )
b. write the area ( a ) as a function of ( l )
c. find the dimensions that produce the greatest area
d. describe in words the significance of the answer to part (c)

Explanation:

Step1: Solve part (a)

We know that the sum of length \( l \) and width \( w \) is 240 feet, so the equation is \( l + w=240 \). To write \( w \) as a function of \( l \), we solve for \( w \) by subtracting \( l \) from both sides of the equation.
\( w = 240 - l \)

Step2: Solve part (b)

The area \( A \) of a rectangle is given by the formula \( A=l\times w \). We already found that \( w = 240 - l \), so we substitute \( w \) in the area formula.
\( A(l)=l\times(240 - l)=240l - l^{2} \)

Step3: Solve part (c)

The area function \( A(l)=-l^{2}+240l \) is a quadratic function in the form \( y = ax^{2}+bx + c \), where \( a=- 1 \), \( b = 240 \), and \( c = 0 \). For a quadratic function, the vertex (which gives the maximum value when \( a<0 \)) has its \( x \)-coordinate (in our case, the value of \( l \)) given by \( l=-\frac{b}{2a} \).
Substitute \( a=-1 \) and \( b = 240 \) into the formula:
\( l=-\frac{240}{2\times(-1)}=\frac{- 240}{-2}=120 \)
Now that we have \( l = 120 \) feet, we find \( w \) using the formula from part (a): \( w=240 - l=240 - 120 = 120 \) feet.

Step4: Solve part (d)

The answer to part (c) (a square with side length 120 feet) is significant because among all rectangles with a given perimeter (here, the sum of length and width is 240, so the perimeter would be \( 2(l + w)=480 \) feet), a square (a special case of a rectangle) has the maximum area. This is a well - known result in geometry and optimization, where for a fixed perimeter, the rectangle with equal length and width (a square) maximizes the area.

Answer:

a. \( w(l)=240 - l \)

b. \( A(l)=240l - l^{2} \)

c. Length \( l = 120 \) feet and width \( w = 120 \) feet (a square with side length 120 feet)

d. Among rectangles with a fixed sum of length and width (or fixed perimeter), a square (rectangle with equal length and width) has the maximum area.