Answer:
Smallest perimeter = 19.6 units
Largest perimeter for area of 24 u2 = 50 units (assuming smallest side = 1 unit)
Largest perimeter for area of 100 u2 = 202 units (assuming smallest side = 1 unit)
Step-by-step explanation:
The area of a rectangle is:
Area = length * width
If the area is 24 u2, we have that:
length * width = 24 -> length = 24/width
Then the perimeter is:
P = 2*length + 2*width = 48/width + 2*width
To find the smallest perimeter, we can take the derivative and make it equal to zero to find the width that gives the smallest perimeter:
dP/dwidth = -48/width^2 + 2 = 0
width^2 = 48/2 = 24
width = 4.9
so the length is 24/width = 24/4.9 = 4.9
and the perimeter is 2*4.9 + 2*4.9 = 19.6 units
To find the largest perimeter, we need to assume the smallest value that a side can have. Assuming that this value is 1 unit, we have that:
Area = length * 1 = 24
length = 24 units
Perimeter = 2*24 + 2*1 = 50 units
If the area is 100 u2, the largest area, again assuming smallest value for side equal 1 unit, is:
Area = length * 1 = 100
length = 100 units
Perimeter = 2*100 + 2*1 = 202 units
