Answer:
Length of the rectangle is either 0.5 units or 1.5 units.
Step-by-step explanation:
Given:
Perimeter of a unit square is equal to the perimeter of a rectangle.
Area of rectangle is 75 % of the square's area.
Unit square means a square of side length 1 unit.
So, perimeter of a unit square,[tex]P_{s}[/tex], is sum of all the 4 sides and is equal to:
[tex]P_{s}=4\times 1=4[/tex]
Therefore, perimeter of the rectangle is, [tex]P_{r}=4[/tex] units.
Now, perimeter of rectangle of length [tex]l[/tex] and width [tex]b[/tex] is given as:
[tex]P_{r}=2(l+b)[/tex]
Therefore,
[tex]2(l+b)=4\\l+b=\frac{4}{2}\\l+b= 2[/tex] --------1
Again, area of a unit square is 1 square units.
So, area of rectangle 0.75 of 1 which is 0.75 square units.
Or, [tex]lb=0.75\\b=\frac{0.75}{l}[/tex]--------2
Plug in [tex]b=\frac{0.75}{l}[/tex] in equation 1, we get
[tex]l+\frac{0.75}{l}=2\\l^{2}+0.75=2l\\l^{2}-2l+0.75=0\\(l-0.5)(l-1.5)=0\\l=0.5\textrm{ or } l=1.5[/tex]
[tex]l=0.5 \textrm{ or }l= 1.5[/tex]
Therefore, the length of the rectangle is 0.5 units or 1.5 units.
