Question

A unit square and a rectangle have the same perimeter. What is the length of the rectangle if its area is 75% of the square's area?

Answer 1

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,$P_{s}$, is sum of all the 4 sides and is equal to:

$P_{s}=4\times 1=4$

Therefore, perimeter of the rectangle is, $P_{r}=4$ units.

Now, perimeter of rectangle of length $l$ and width $b$ is given as:

$P_{r}=2(l+b)$

Therefore,

$2(l+b)=4\\l+b=\frac{4}{2}\\l+b= 2$ --------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, $lb=0.75\\b=\frac{0.75}{l}$--------2

Plug in $b=\frac{0.75}{l}$ in equation 1, we get

$l+\frac{0.75}{l}=2\\l^{2}+0.75=2l\\l^{2}-2l+0.75=0\\(l-0.5)(l-1.5)=0\\l=0.5\textrm{ or } l=1.5$

$l=0.5 \textrm{ or }l= 1.5$

Therefore, the length of the rectangle is 0.5 units or 1.5 units.