Math 1321
Week 10 Lab Worksheet
Due Thursday 03/28
1. Designing a swimming pool For this project, we will consider the task of designing a
swimming pool with a prescribed volume, while minimizing the cost of installation. To
this end, we will consider a pool such as the one depicted below. Furthermore we will
make the following assumptions about digging and installation of a pool.
• The sides of the pool will be made of a thinner plaster/concrete mix at a cost of
$5.00 per square foot.
• The bottom of the pool will be made of a thicker plaster/concrete mix at a cost of
$10.00 per square foot.
• The deep-end or flat part of the bottom of the pool, remains exactly 25% of the
length of the pool.
• The shallow-end dives off toward the deep-end immediately as shown in the above
image.
• Excavation costs are typically $5.00 per cubic foot.
• All other labor costs will be ignored for simplicity.
(a) Find an equation which describes the cost laying the plaster/concrete mix for the
3 sides of the pool.
(b) Find an equation which describes the cost of laying the bottom layer of plaster/concrete mix of the pool. Hint: Remember the Pythagorean Theorem and that
the deep end is always 25% the length of the entire pool.
(c) Find an equation which describes the volume V (x, y, z) of the pool for arbitrary
dimensions x, y, and z.
(d) Find an equation which describes the cost C(x, y, z) of installing the pool, including
excavation costs.
(e) Set up a constrained cost minimization procedure using Lagrange multipliers by
requiring the volume V to be the same as in the image above, while minimizing the
cost function C(x, y, z). (Do not solve the minimization problem).
Solution:
(a) Letting x denote the length of the pool, y denote the width of the pool, and
z denote the depth of the deep end of the pool, we need to find an expression
for the area of the 3 sides of a pool with dimensions x, y, and z. The two walls
which span from the shallow end to the deep end may be split into two easier
shapes, namely, a triangle and a rectangle. Since the deep end is exactly 25% of
the length of the pool, we find the area of this square to (0.25)xzf t2 . Thus the
remaining part of the wall, the triangle, has area 12 · (0.75)xzf t2 . So the total
area of one of these walls is given by Asides = 2((0.25)xz + 12 (0.75)xz)f t2 . The
area of the back wall is found to be Aback = yzf t2 . Thus we are able to find
the total cost of installing the pool walls by multiplying this total area by the
cost per area. We find Cwalls (x, y, z) = 2((0.25)xz + 21 (0.75)xz)(5.00) + yz(5.00)
USD.
(b) Similar to part (a), we compute the area of the bottom of the pool- the bottom
of the deep end and the sloped section
p up to the shallow end. We find the
area to be Abottom = (0.25)xy + y ((0.75)x)2 + z 2 , and thus the total cost
ofpinstalling the bottom of the pool is given by Cbottom (x, y, z) = ((0.25)xy +
y ((0.75)x)2 + z 2 )(10.00) USD.
(c) The volume of the pool can be split into finding the volume of two easier shapes,
a rectangular cube, and a triangular wedge or prism. The volume of the cube
(the deep end of the pool) is given by Vcube = (0.25)xyzf t3 while the volume of
the prism (the shallowing slope) is given by Vwedge = 21 (0.75)xzyf t3 . Thus the
volume of the pool for arbitrary dimensions x, y, z is found to be V (x, y, z) =
(0.25)xyz + 21 (0.75)xyzf t3 .
(d) The overall cost of installing the pool will be found by combining the cost of
laying all of the plaster/concrete mix and the excavation costs. We have already computed the plaster/concrete costs, so we need to find the excavation
costs. We are given that excavation typically costs $5.00 per cubic foot, and
since we know the volume of the pool, we know how much we need to excavate.
Therefore the cost of installing the pool is given by C(x, y, z) = Cwalls (x, y, z) +
Cbottom (x, y, z) + V (x, y, z) · (5.00) = 2((0.25)xz + 12 (0.75)xz)(5.00) + yz(5.00) +
p
((0.25)xy+y ((0.75)x)2 + z 2 )(10.00)+((0.25)xyz+ 12 (0.75)xyz)(5.00) = 6.25xz+
√
5.00yz + 2.5xy + 10.00y .5625x2 + z 2 + 3.125xyz USD.
(e) To set up the constrained minimization problem we need to set ∇C(x, y, z) =
λ∇V (x, y, z) for some λ and require V (x, y, z) = 15000f t3 . Thus we need to
solve the nonlinear system
5.625yx
6.25z + 2.5y + √
+ 3.125yz
.5625x2 + z 2
√
5.00z + 2.5x + 10.00 .5625x2 + z 2 + 3.125xz
10.00yz
6.25x + 5.00y + √
+ 3.125xy
.5625x2 + z 2
1
(0.25)xyz + (0.75)xyz
2
= λ0.625yz
= λ0.625xz
= λ0.625xy
= 15000