Question

Of all rectangles with area 100, which one has the minimum perimeter? Let P and w be the perimeter and width, respectively, of the rectangle. Write the objective function in terms of P and w. Assume that the width is less than the length if the dimensions are unequal. P= (Type an expression.) The interval of interest of the objective function is (Simplify your answer. Type your answer in interval notation.) Of all rectangles with area 100, the one with dimensions has the minimum perimeter. (Simplify your answers. Use a comma to separate answers as needed.)

Answer 1

anea of the mock Let A be the rectangle, L =length, w=width. Then, acconding to the given problem. A = LW=100 Now, if P be the perimeten of the nectangle then P= 2 (L+ W) => P. = 2 (100 otw) [by 0] This is the required objective function for the perimeton P in terms of w. P (W) 200 taw Now, interval of interest means actually domain of the function at which the function is defined Hene, PCW) is defined for two i.e, intenval of (0,0) of interest is.

for minimum we must have dP = 0 perimeter. dw -> - 200 +2=0 W2 -2 200 = 2 w2 2 -> w - 100 = w + 10 Since, wo W =10. Again, d²p 400 dwa 43 dp - 400 dwz d/ W210 1000 so, P (W) is minimum at w=10 Then 1= 100 100 - 10 :. Perimeten is minimum when L=W= 10 units le length = 10 units breadth=10 units 2 This is the required dimension. for that minimum perimeters 2 (10+10) = 40 units And

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