Question

boundary of a field is a right triangle with a straight stream along its hypotenuse and with

fences along its other two sides. Find the dimensions of the field with maximum area that can be

enclosed using 500 meters of fence. Give answer in feet​

Answer 1

Given : boundary of a field is a right triangle with a straight stream along its hypotenuse and with  fences along its other two sides

To find : the dimensions of the field with maximum area that can be

enclosed using 500 meters of fence

Solution:

Fence along perpendicular Sides

Let say perpendicular Sides B & H

B + H  = 500

=> H = 500 - B

Area of Triangle = (1/2) * B * H

=> A  = (1/2) * B  (500 - B)

=> A = (1/2) ( 500B - B² )

dA/dB  = (1/2) ( 500 - 2B)

dA/dB  = 0

=>  (1/2) ( 500 - 2B) = 0

=> B = 250

d²A/dB² = (1/2)(0 - 2)

=> d²A/dB² = - 1  < 0

Hence area is maximum for  B = 250  

H = 500 - B = 500 - 250 = 250

Hypotenuse = √(250)² + (250)²  = 250√2

dimensions of the field with maximum area  = 250 , 250 , 250√2    m

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