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 License Agreement At The Make It Real Learning Company, our goal is to provide quality instructional materials at a price that even an entry-level teacher can afford. By complying with this license agreement, you help us reach that goal. We thank you for your support. Acceptable Use As a paid subscriber, you may make hard copies of this worksheet for use in any classes that you teach. This includes traditional teacher-student classes as well as professional development workshops that you lead. 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