Find the dimensions of the rectangle with area 256 square inches that has minimum perimeter, and then find the minimum perimeter. 1. Dimensions: equation editorEquation Editor 2. Minimum perimeter: equation editorEquation Editor Enter your result for the dimensions as a comma separated list of two numbers. Do not include the units.

Answer:Dimensions: [tex]A=a\cdot b=256[/tex]Perimiter: [tex]P=2a+2b[/tex]Minimum perimeter: [16,16]Step-by-step explanation:This is a problem of optimization with constraints.We can define the rectangle with two sides of size "a" and two sides of size "b".The area of the rectangle can be defined then as:[tex]A=a\cdot b=256[/tex]This is the constraint.To simplify and as we have only one constraint and two variables, we can express a in function of b as:[tex]b=\frac{256}{a}[/tex]The function we want to optimize is the diameter.We can express the diameter as:[tex]P=2a+2b=2a+2*\frac{256}{a}[/tex]To optimize we can derive the function and equal to zero.[tex]dP/da=2+2\cdot (-1)\cdot\frac{256}{a^2}=0\\\\\frac{512}{a^2}=2\\\\a=\sqrt{512/2}= \sqrt {256} =16\\\\b=256/a=256/16=16[/tex]The minimum perimiter happens when both sides are of size 16 (a square).