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As a result of complaints by the staff about noise, the coffee and recreation area for student interns at OHaganBooks will now be in a 512 square foot rectangular area in the headquarter's basement against the southern wall. (The specified area was arrived at in complex negotiations between the student intern representative and management.) The construction of the partition will cost $16 per foot for the north wall and $4 per foot for the east and west walls. What are the dimensions of the cheapest recreation area that can be made?

Respuesta :

Answer:

The dimensions are 32 ft by 16 ft

Step-by-step explanation:

Area  of the coffee and recreation room=512 square foot

LB=512

L=512/B

Perimeter of the Room = Perimeter of north wall+Perimeter of east wall+ Perimeter of west wall =L+2B (West and East are opposite)

Cost of the Perimeter=16L+4(2B)

[tex]C(B)=16(\frac{512}{B})+8B\\C(B)=\frac{8192+8B^2}{B}[/tex]

To minimize cost, first, we take the derivative of C(B)

Using quotient rule

[tex]C^{'}(B)=\frac{8B^2-8192}{B^2}[/tex]

Setting the derivative equal to zero

[tex]8B^2-8192=0\\8B^2=8192\\B^2=1024\\B=32 ft[/tex]

[tex]L=\frac{512}{B}=\frac{512}{32}=16ft[/tex]

The dimension of the cheapest recreation area will be 32 ft by 16 ft

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