Swimming Pool Design

Swimming Pool Design
Working with Perimeters and Areas
I
n 2006, Sahara Pools and Spas offered a basic in-ground swimming pool for $15,950. The pool could
be of any shape but the perimeter of the pool had to be 75 feet or less and the surface area had to be
350 square feet or less. For customers wanting a larger pool, each extra foot of perimeter cost $95 and
each extra square foot of area cost $8.50 (Source: Sahara Pools and Spas contract).
1. One simple pool design is a rectangle joined with a semicircle (see figure). Determine the perimeter and
the area of the pool.
The area of a rectangle is the product of its length and width. That is, A = lw . The length of the
rectangle is 21 feet and the width of the rectangle is 14 feet.
A = 21 (14 )
= 294 square feet
The area of a circle is the product of π and the square of the radius. That is, A = π r 2 . So the area of a
semicircle is A = 12 π r 2 . (Recall π ≈ 3.141 .) Since the diameter of the semicircle is 14 feet, its radius is
7 feet.
2
A = 12 π ( 7 )
≈ 76.97 sq ft
≈ 77 sq ft
Combining both areas, we determine the total area of the pool is about 371 square feet.
The circumference of a circle is C = 2π r . Therefore, the distance around the outside of the semicircle is
C = π r . The perimeter of the pool is given by
p = 21 + 14 + 21 + π (7 )
≈ 78 feet
2. The pool in (1) is larger than the basic pool covered by the $15,950 price. What will be the total cost of
the pool?
We need to determine the amount of extra area and extra perimeter.
Extra perimeter = Total perimeter − 75
Extra area = Total area − 350
= 78 − 75
= 371 − 350
= 3 feet
= 21 square feet
Since each extra square foot of area costs $8.50 and each extra foot of perimeter costs $95, the extra
cost is
C = 21 ( 8.50 ) + 95 ( 3)
= $463.50
Adding this cost to the base price of the pool yields $16,413.50.
3. Is it possible to make a rectangular pool that has a perimeter of exactly 75 feet and an area of exactly
350 square feet? Complete the given table as a part of the problem solving process.
Length
5
Width
Area
Perimeter
75
10
75
15
75
20
75
25
75
30
75
35
75
Yes. A pool with length 20 feet and width 17.5 feet has an area of 350 square feet and a perimeter of
75 square feet.
4. Is it possible to make a circular pool with a circumference of 75 feet and an area of 350 square feet?
The formula for the circumference of a pool is C = 2π r . Since we know the circumference of the pool is
75 feet, we can find the radius r.
75 = 2π r
75
r =
2π
r ≈ 11.94
75
The formula for the area of a circle is A = π r 2 . We calculate the area of the circle with radius r =
.
2π
2
75
A = π ⎛⎜ ⎞⎟
⎝ 2π ⎠
≈ 447.6 squre feet
No. It is not possible to make a circular pool with circumference 75 feet and area 350 square feet.
Swimming Pool Design
Working with Perimeters and Areas
I
n 2006, Sahara Pools and Spas offered a basic in-ground swimming pool for $15,950. The pool could
be of any shape but the perimeter of the pool had to be 75 feet or less and the surface area had to be
350 square feet or less. For customers wanting a larger pool, each extra foot of perimeter cost $95 and
each extra square foot of area cost $8.50 (Source: Sahara Pools and Spas contract).
1. One simple pool design is a rectangle joined with a semicircle (see figure). Determine the perimeter and
the area of the pool.
The area of a rectangle is the product of its length and width. That is, A = lw . The length of the
rectangle is 21 feet and the width of the rectangle is 14 feet.
A = 21 (14 )
= 294 square feet
The area of a circle is the product of π and the square of the radius. That is, A = π r 2 . So the area of a
semicircle is A = 12 π r 2 . (Recall π ≈ 3.141 .) Since the diameter of the semicircle is 14 feet, its radius is
7 feet.
2
A = 12 π ( 7 )
≈ 76.97 sq ft
≈ 77 sq ft
Combining both areas, we determine the total area of the pool is about 371 square feet.
The circumference of a circle is C = 2π r . Therefore, the distance around the outside of the semicircle is
C = π r . The perimeter of the pool is given by
p = 21 + 14 + 21 + π (7 )
≈ 78 feet
2. The pool in (1) is larger than the basic pool covered by the $15,950 price. What will be the total cost of
the pool?
We need to determine the amount of extra area and extra perimeter.
Extra perimeter = Total perimeter − 75
Extra area = Total area − 350
= 78 − 75
= 371 − 350
= 3 feet
= 21 square feet
Since each extra square foot of area costs $8.50 and each extra foot of perimeter costs $95, the extra
cost is
C = 21 ( 8.50 ) + 95 ( 3)
= $463.50
Adding this cost to the base price of the pool yields $16,413.50.
3. Is it possible to make a rectangular pool that has a perimeter of exactly 75 feet and an area of exactly
350 square feet? Complete the given table as a part of the problem solving process.
Length
5
Width
32.5
Area
162.5
Perimeter
75
10
27.5
275
75
15
22.5
337.5
75
20
17.5
350
75
25
12.5
312.5
75
30
7.5
225
75
35
2.5
87.5
75
Yes. A pool with length 20 feet and width 17.5 feet has an area of 350 square feet and a perimeter of
75 square feet.
4. Is it possible to make a circular pool with a circumference of 75 feet and an area of 350 square feet?
The formula for the circumference of a pool is C = 2π r . Since we know the circumference of the pool is
75 feet, we can find the radius r.
75 = 2π r
75
r =
2π
r ≈ 11.94
75
The formula for the area of a circle is A = π r 2 . We calculate the area of the circle with radius r =
.
2π
2
75
A = π ⎛⎜ ⎞⎟
⎝ 2π ⎠
≈ 447.6 squre feet
No. It is not possible to make a circular pool with circumference 75 feet and area 350 square feet.
Worksheet Title
Keywords
Swimming Pool Design: Working with Perimeters and Areas
Pool, construction, swimming, price, area, perimeter, geometry, circle, rectangle
NCTM Standard
X
X
X
X
Grade Band
Data Type
Filename: m3001
X
X
Words
Content Standards
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
PreK − 2
3−5
6−8
9 − 12
X
X
X
X
Process Standards
Problem Solving
Reasoning and Proof
Communication
Connections
Representations
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