Question

A rectangular building is being designed to minimize heat loss. The east and west walls lose

heat at a rate of 10 units/m2 per day, the north and south walls lose heat at a rate of 8 units/m2

per day, and the floor loses heat at a rate of 8 units/m2 per day. Each wall must be at least 30

m long, the height must be at least 4 m and the volume exactly 4000 m3:

(a) Find and sketch the domain of the heat loss as a function of the lengths of the sides.

(b) Find the dimensions that minimize heat loss. (Check both the critical points and the points

on the boundary of the domain.)

(c) Could you design a building with even less heat loss if the restrictions on the lengths of the

walls were removed.

Answer 1

Check the explanation

Explanation:

Let x be the length of the north and south walls, y the length of the east and

west walls, and z the height of the building. The heat loss is given by

h = 10(2yz) + 8(2xz) + 1(xy) + 5(xy) = 6xy + 16xz + 20yz.

The volume is 4000 m3, so xyz = 4000, and we substitute z = 4000/(xy) to obtain the

heat loss function

h(x, y) = 6xy + 80, 000/x+ 64, 000/y.

a) Since 4000/(xy)≥4, xy≤1000, i.e., y ≤1000/x. Also x ≥ 30 and y≥ 30, so the

domain of h is D= {(x, y) : x ≥30, 30≤ y≤ 1000/x}.

is the region bounded from below by the horizontal line segment from (30

,30)to ( 100/3 , 30) (let us call this line ), from the right by the portion of the hyperbola y = 1000/x from (30, 100/3 ) to ( 100/3) (we call this curve ) and from the left by the vertical line segment from (30, 30) to (30, 100/3 ) (denote this by )

b) Kindly check the solution to question B in the attached image below.

h is h(30, 30) = 10, 200 and the dimensions of the

building that minimize heat loss are walls 30 m in length and height 4000

/302 =

40

/9 = 4.44 m.

(c) From part (b), the only critical point of

h, which gives a local and absolute minimum x ≈ 25.54 m, y = 20.43 m, z ≈ 4000 /(25.54)(20.43) ≈ 7.6

I hope it helped you.

is approximately

h(25.54, 20.43) ≈ 9396. So a building of volume 4000 m3 with dimensions.