Find the dimensions of a rectangle divided into two equal rectangles of maximum area that has total perimeter (including the mid

Question

2 years ago

11

dle divider) of 120 feet.

1 answer:

lilavasa [31] · Answer 1 · 2023-03-18T00:06:54+03:00

The dimensions for this rectangle are b= 30 ft and h=20 ft.

<h3>Perimeter</h3>

The perimeter of a geometric figure is the sum of its sides.

The question gives:

Therefore for the perimeter, you have

P= $\frac{b}{2} +\frac{b}{2}$ + h+ h +h

P=2b+3h

2b+3h=120 (1)

From equation 1, you can write:

2b=120-3h

b= $\frac{120-3h}{2}$

b= 60 - $\frac{3h}{2}$ (2)

The area for the rectangle is given by A= bh. Therefore, by replacing (2) in the formula for rectangle area, you have:

$A= (60-\frac{3h}{2} )* h\\ \\ A= 60h-\frac{3h^2}{2}$

The maximum area will be calculated from the derivation of the previous equation of area.

$\frac{dA}{dh}=60-2*\frac{3h}{2}\\ \\ \frac{dA}{dh}=60-3h$

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 - $\frac{3h}{2}$ ) you can find the base.

b= 60 - $\frac{3*20}{2}$

b=60 - 3*10

b=60-30

b=30

Read more about the First Derivative Test here:

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