Respuesta :
Answer:
the dimensions that will minimize the cost are; 25.2 ft × 25.2 ft × 12.6 ft
Step-by-step explanation:
If the length, width and height of the building are represented by a, b and c, then we have:
Volume = abc
We are given volume as 8000 ft³. Thus;
abc = 8000
Top area = ab
Front area = ac
Back area = ac
End walls area = bc
We are told that cooling costs will amount to $2/ft² for its top, front, and back, and $4/ft² for the two end walls.
Thus, cooling cost equation is;
X = 2(ab) + 2(ac) + 2(ac) + 4bc + 4bc
X = 2ab + 4ac + 4bc
Now, from the volume equation where abc = 8000.
c = 8000/ab
Thus;
X = 2ab + 4a(8000/ab) + 4b(8000/ab)
X = 2ab + 32000/b + 32000/a
To minimize, we will find the a and b derivative and set the equation to zero in both cases.
Thus;
dX/da = 2b - 32000/a² = 0
2b = 32000/a² - - - eq(1)
Also;
dX/db = 2a - 32000/b² = 0
2a = 32000/b² - - - (eq 2)
Making a the subject in eq 2 gives;
a = 32000/2b²
a = 16000/b²
Putting 16000/b² for a in eq (1) gives;
2b = 32000/(16000/b²)²
2b = 32000b⁴/256000000
2b = b⁴/8000
Multiply both sides by 8000 to get;
2b × 8000 = b⁴
16000b = b⁴
Divide both sides by b to get;
16000 = b³
b ≈ 25.2 ft
Putting 25.2 for b in a = 16000/b², we have;
a = 16000/25.2²
a ≈ 25.2
Putting a and b = 25.2 in the volume equation, we have;
25.2 × 25.2 × c = 8000
c = 8000/(25.2 × 25.2)
c = 12.6 ft
Thus, the dimensions that will minimize the cost are; 25.2 ft × 25.2 ft × 12.6 ft
