Mrs. Napholtz's Math Site

Rectangle Problems

If the side of a square is increased by 3 m, its area is 121 m^{2}. Find the length of a side of the original square.

(s + 3)^{2} = 121

s^{2} + 6s + 9 = 121

s^{2} + 6s - 112 = 0

(s + 14)(s - 8) = 0

s = 8 is the only reasonable answer.

The length of a rectangular table is 5 in. more than twice its width. Its area is 1950 in^{2}. Find its dimensions.

Let w = width.

Length = 2w + 5

w(2w + 5) = 1950

2w^{2} + 5w - 1950 = 0

(2w + 65)(w - 30) = 0

w = 30 is the only reasonable solution.

ANS: 30 by 65.

The diagonal of a rectangle is 25 cm, and the length is 24 cm. Find the width of the rectangle.

Using the Pythagorean Theorem we get:

l^{2} + w^{2} = diag^{2}

24^{2} + w^{2} = 25^{2}

576 + w^{2} = 625

w^{2} = 49

w = 7

A photograph is 12 cm longer than it is wide. It is mounted in a frame 5 cm wide. The area of the frame is 620 cm ^{2}. Find the dimensions of the photo.

| Length | Width | Area |

Photo | n + 12 | n | n |

Photo plus frame | n + 22 | n + 10 | n |

Area of the frame = 620.

n^{2} + 32n + 220 - (n^{2}+ 12n) = 620

n^{2} +32n + 220 - n^{2} - 12n = 620

32n + 220 - 12n = 620

20n + 220 = 620

20n = 400

n = 20

Answer: The dimensions of the photo are 20 by 32.

The side of a square has length s. The length of a rectangle is 4 m more than the side of the square and the width of the rectangle is 2 m less than the side of the square. The perimeter of the rectangle is 24 m less than twice the perimeter of the square. Find the dimensions of each figure.

| Length | Width | Perimeter |

Square | n | n | 4n |

Rectangle | n + 4 | n - 2 | 4n + 4 |

The perimeter of the rectangle is 24 less than twice the perimeter of the square.

4n + 4 = 2(4n) - 24

4n + 4 = 8n - 24

4n + 28 = 8n

28 = 4n

n = 7

Answer: The square is 7 by 7. The rectangle is 11 by 5.

The length of a rectangle is 1 cm less than three times the width. If the area of the rectangle is

44 cm^{2}, find the dimensions.

Width = w

Length = 3w - 1

Area = 44

w(3w - 1) = 44

3w^{2} - w = 44

3w^{2} - w - 44 = 0

(3w + 11)(w - 4) = 0

3w + 11 = 0 OR w - 4 = 0

w = -11/3 NOT A VALID DIMENSION OR w = 4.

Answer: Width = 4, Length = 11

A rectangular enclosure with one partition is to be bounded by a wall and 30 m of fencing, as shown in the diagram. What should the dimensions be to form a rectangle whose area is 48 m^{2}

The fence consists of 4 parts: 3 parallel sections, each of length x, and 1 section, parallel to the wall, whose length must be 30 - 3x.

The dimensions of the enclosure are: width = x, length = 30 - 3x.

Area = 48.

x(30 - 3x) = 48

30x - 3x^{2} = 48 Add 3x^{2} and subtract 30x

0 = 3x^{2} - 30x + 48 Factor out 3.

0 = 3(x^{2} - 10x + 16)

0 = 3(x - 2)(x - 8)

x - 2 = 0 OR x = 8 = 0.

x = 2 OR x = 8.

If x = 2, then length = 30 - 3(2) = 24. Answer: 2 by 24

If x = 8, then length = 30 - 3(8) = 6. Answer: 8 by 6.

Both are valid configurations.

The length of a rectangle is 6 m more than twice the width. If the area of the rectangle is 140 m^{2}, find the dimensions.

The dimensions of the rectangle are w and 2w + 6.

Area = 140.

w(2w + 6) = 140

2w^{2} + 6w = 140

2w^{2} + 6w - 140 = 0

2(w^{2} + 3w - 70 = 0

2(w + 10)(w - 7) = 0

w + 10 = 0 OR w - 7 = 0

w = -10 Not a valid dimension. OR w = 7.

Answer: width = 7 and length = 2(7) + 6 = 20.

A rectangular pool is 8 m longer than it is wide. A walkway 2 m wide surrounds the pool. What are the dimensions of the pool if the area of the walkway is 176 m^{2}?

| Length | Width | Area |

Pool | n + 8 | n | n |

Pool and Walkway | n + 12 | n + 4 | n |

Area of the walkway = 176.

n^{2} + 16n + 96 - (n^{2}+ 8n) = 176

n^{2} + 16n + 96 - n^{2} - 8n = 176

16n + 96 - 8n = 176

8n + 96 = 176

8n = 80

n = 10

Answer: The dimensions of the pool are 10 by 18.

A rectangular garden plot is 10 m longer than it is wide. If the length is increased by 1 m and the width decreased by 2 m, the area is decreased by 42 m^{2}. What are the original dimensions?

The new area is 42 less than the old area.

(w + 11)(w - 2) = w(w + 10) - 42

w^{2} -2w + 11w - 22 = w^{2} + 10w - 42

9w - 22 = 10w - 42 Add -10w

-w - 22 = -42 Add 22

-w = -20

w = 20

Answer: The original dimensions were 20 by 30

A poster is 25 cm taller than it is wide. It is mounted on a piece of cardboard so that there is a 5 cm border on all sides. If the area of the border alone is 1350 cm^{2}, what are the dimensions of the poster?

Outer Area - Inner Area = Border = 1350

(x + 35)(x + 10) - x(x + 25) = 1350

x^{2} + 45x + 350 - x^{2} - 25x = 1350

20x + 350 = 1350

20x = 1000

x = 50

ANSWER: 50 by 75

A rectangle is twice as long as it is wide. If its length and width are both decreased by 4 cm, its area is decreased by 164 cm^{2}. Find its original dimensions.

The new area is 164 less than the old area.

(2w - 4)(w - 4) = 2w^{2} - 164

2w^{2} - 8w - 4w + 16 = 2w^{2} - 164

-12w + 16 = -164 Add -16

-12w = -180

w = 15

Answer: Original dimensions: 15 cm by 30 cm.

A rectangle is three times as long as it is wide. If its length and width are both increased by 3 m, its area is increased by 81 m^{2}. Find its original dimensions.

The new area is 81 more than the old area.

(3w + 3)(w + 3) = 3w^{2} + 81

3w^{2} + 9w + 3w + 9 = 3w^{2} + 81

12w + 9 = 81 Add -9

12w = 72

w = 6

Answer: Original dimensions: 6 m by 18 m.

A rectangle is 8 cm longer than it is wide. If its length and width are both increased by 2 cm, its area is increased by 68 cm^{2}. Find its original dimensions.

The new area is 68 more than the old area.

(w + 10)(w + 2) = w^{2} + 8w + 68

w^{2} + 2w + 10w + 20 = w^{2} + 8w + 68

12w + 20 = 8w + 68 Add - 8w

4w + 20 = 68

4w = 48

w = 12

Answer: Original dimensions: 12 cm by 20 cm.

A carpenter builds two decks with the same area. One deck is a square and the other is a rectangle which is 4 m longer than the square but 3 m less wide than the square. What is the area of each deck?

| Length | Width | Area |

Square | n | n | n |

Rectangle | n + 4 | n - 3 | n |

Since the areas are equal, we get

n^{2} = n^{2} + n - 12

0 = n - 12

12 = n

Answer:

Area of square deck = 12^{2} = 144 m^{2}

Area of rectangular deck = (12 + 4)(12 - 3) = (16)(9) = 144 m^{2}