Optimization Project
8-3 Optimization
Project
Kyla Semones
Problem
• After returning home from Spring Break in the Bahamas, Sarah realizes she can’t live without the beach. To satisfy her needs she decides to add an addition to her house
(not directly connected). This will include a beach like atmosphere.
– Sarah needs help finding the dimensions of the room using the minimum amount of material required to build this addition which has a volume of 3,000ft 3 . Her new beach will be added onto her patio so all we have to worry about is the ceiling and walls. Find the dimensions of the room and give the minimum amount of material needed.
Diagram
Solution
• Solve for a variable using volume given
– 3000=.5x
2 y
– Y=3000/(.5x
2 ) or y=6000x -2
• Create Equation for Surface Area of
Box using values shown in diagram
– A(x)=.5xy+xy+.5x
2
– A(x)=xy+xy+x 2
– A(x)=2xy+x 2
Solution
• Substitute the y value found in the first step.
– A(x)=2x(6000x -2 )+x 2
– A(x)=x 2 +12000x -1
• Find the derivative of the surface area function
– A’(x)= 2x-12000x -2
• Set equal to zero and solve for x.
– 0=2x-12000/(x 2 )
– X=18.1712
Solution
• Use original volume equation and x value to solve for y.
– 3000=.5(18.1712) 2 y
– Y=18.1712
Answer the Question
Given the diagram below, and the fact that x and y are equal to 18.1712. The dimensions of the new beach are,
18.1712 x 18.1712 x 9.0856.
1
st
Derivative
• Find the derivative of surface area function
– A’(x)= 2x-12000x -2
• Set equal to zero and solve for x.
– 0=2x-12000/(x 2 )
– X=18.1712
• Plug in 18 and 19 and compare the slopes.
– A’(18)=2(18)-12000(18) -2 A’(18)=-1.0370
– A’(19)=2(19)-12000(19) -2 A’(19)= 4.7590
1
st
Derivative
The slope changes from negative to positive. The graph goes from decreasing to increasing therefore minimum must occur between 18 and
19. This minimum occurs at 18.1712.
2
nd
Derivative
• Find the second derivative of surface area function
– A’’(x)= 2+24000x -1
• Substitute the x value found previously
– A’’(18.1712)= 2+24000(18.1712) -1
– A’’=1322.77
2
nd
Derivative
When x=18.1712 the second derivative is positive. This means the graph is concave up and there is a minimum.
Problem
Now Sarah wants to know the minimum amount of material needed to build the walls and ceiling.
Solution
• X= 18.1712
• Y= 18.1712
• Substitute x and y into the surface area equation.
– A(x)=2xy+x 2
– A= 2(18.1712)(18.1712)+(18.1712) 2
– A= 990.5775ft
2
Recap
• The dimensions of the room are
18.1712 x 18.1712 x 9.0856.
• The minimum amount of material needed is 990.5775ft
2
