Homework SourseA mason has enough brick to build a 46 foot wall The homeown
A mason has enough brick to build a 46 foot wall. The homeowners want to use the wall to enclose an outdoor patio. The patio will be along the side of their house and will include a 4-foot opening for a door. What dimensions will maximize the are of the patio? a. identify the output variable to be optimized and the input variable(s) b. sketch and label a diagram c. write a model for the output variable in terms of one input variable. d. answer the question posed.
Solution
So there are two sides that are equal length and one more side that has a 4-foot hole in it. The output variable is AREA, the inputs are LENGTH and WIDTH. The area is just A=xy The perimeter is P=2x+(y-4)=46 y=50-2x A=x*(50-2x) dA/dx=0 for optimization dA/dx=x(-2)+(50-2x)=0 50-4x=0 x=12.5ft y=50-2(12.5)= 25ft So the dimensions are 12.5ft by 25ft, with the 4ft door cut \"somewhere\" in the middle of the 25ft-wall.
