The dimensions for this rectangle are b= 30 ft and h=20 ft.
Perimeter
The perimeter of a geometric figure is the sum of its sides.
The question gives:
- the geometric figure - rectangle
- the total perimeter - 120 ft
Therefore for the perimeter, you have
P= [tex]\frac{b}{2} +\frac{b}{2}[/tex]+ h+ h +h
P=2b+3h
2b+3h=120 (1)
From equation 1, you can write:
2b=120-3h
b=[tex]\frac{120-3h}{2}[/tex]
b= 60 - [tex]\frac{3h}{2}[/tex] (2)
The area for the rectangle is given by A= bh. Therefore, by replacing (2) in the formula for rectangle area, you have:
[tex]A= (60-\frac{3h}{2} )* h\\ \\ A= 60h-\frac{3h^2}{2}[/tex]
The maximum area will be calculated from the derivation of the previous equation of area.
[tex]\frac{dA}{dh}=60-2*\frac{3h}{2}\\ \\ \frac{dA}{dh}=60-3h[/tex]
The maximum area can be found when the first derivative is equal to zero. Thus,
60-3h=0
-3h=-60 *(-1)
3h=60
h=20
Now, you know the height and from equation 2 (b= 60 - [tex]\frac{3h}{2}[/tex] ) you can find the base.
b= 60 -[tex]\frac{3*20}{2}[/tex]
b=60 - 3*10
b=60-30
b=30
Read more about the First Derivative Test here:
https://brainly.com/question/6097697
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