A rectangular microprocessor chip is designed to have an area of 121mm^2.What must its dimensions be if its perimetar is to be a minimum?
for any given rectangle maximum area for a given perimeter (or minimum perimeter for given area) is a square.
so, we want a square of side 11mm
To find the dimensions of the rectangular microprocessor chip with the minimum perimeter given a fixed area, we can use the concept of calculus optimization.
Let's denote the length of the rectangle as "l" and the width as "w". We are given that the area of the rectangle is 121 mm², so we have the equation:
l * w = 121 (Equation 1)
We want to minimize the perimeter of the rectangle, which is given by the formula:
P = 2l + 2w
To find the minimum perimeter, we need to find the values of l and w that satisfy Equation 1 while minimizing P. Here's how we can do it step by step:
Step 1: Express the perimeter formula in terms of a single variable.
Since we have the equation l * w = 121 (Equation 1), we can express one variable in terms of the other. Let's solve for l:
l = 121 / w
Substitute this value of l into the perimeter formula:
P = 2(121 / w) + 2w
P = 242 / w + 2w
Step 2: Differentiate the perimeter formula.
Differentiate P with respect to w to find the critical points. We want to find where the derivative equals zero or does not exist:
dP/dw = -242 / w² + 2
Step 3: Solve for the critical points.
Set the derivative equal to zero and solve for w:
-242 / w² + 2 = 0
-242 + 2w² = 0
2w² = 242
w² = 121
w = ±√121
w = ±11
Since dimensions cannot be negative, we take the positive value: w = 11.
Step 4: Find the corresponding value of l.
Use Equation 1 to find the corresponding value of l:
l = 121 / w
l = 121 / 11
l = 11
So, the dimensions of the microprocessor chip that minimize the perimeter while having an area of 121 mm² are l = 11 mm and w = 11 mm.