WORD PROBLEMS ON AREA OF SHADED REGION

Problem 1 :

The diagram below shows the rectangular pen in which Gomer confines his wolverines and badgers. As a special treat, Gomer is going to cover the badgers' area of the pen with Astroturf. Assuming that Astroturf costs $1.25 per square foot, how much with this project cost?

A. $1500    B. $9000    C. $4500    D. $3000

Solution :

Area of the green shaded region = (1/2) x base x height

base = 40 yards and height = 20 yards

= (1/2) x 40 x 20

= 400 square yards

Now square yards to be converted as square feet

1 yard = 3 feet

400 square yards = 400 x 3 x 3 square feet

= 3600 square feet

Cost per square feet = $1.25

Required cost = 3600 (1.25)

= $4500

So, option C is correct.

Problem 2 :

The diagram below shows the plan for a new parking lot at the South westdale Mall. It is estimated that such construction costs $12 per square yard. What will be the total cost for the parking lot?

A. $2,073,600         B. $691,200

C. $6,220,800          D. $518,400

Solution :

The given shape is a parallelogram.

Area of parallelogram = base x height

base = 960 feet and height = 540 feet

1 yard = 3 feet

Converting into yards, we get

base = 960/3 ==> 320 yards

height = 540/3 ==> 180 yards

= 320(180)

= 57600 square yards

Cost = $12

Required cost = 57600 (12)

= 691200

So, option B is correct.

Problem 3 :

The diagram below shows a circular pizza whose diameter is 8 inches, situated in a square box whose side length is 10 inches. How much of the box is "empty?"

Solution :

To find the empty area, we have to subtract area of circular portion from area of square box.

Area of square - area of circular

Area of square = a x a

Area of circle = πr²

= 10 x 10 - 3.14(4)²

= 100 - 3.14(16)

= 100 - 50.24

= 49.76 square inches.

Problem 4 :

The diagram below shows one exterior wall of a house. The wall has a door opening that measures 3 feet by 7 feet, and two window openings that each measure 6 feet by 4 feet. The wall, but not the openings, will be covered with siding material that costs $1.50 per square foot. Find the total cost of the siding material.

Solution :

Area of top triangle = (1/2) x base x height 

= (1/2) x 24 x 6

= 72 square feet

Area of door = 3 x 7

= 21 square feet

Area of window opening = 2 x 6 x 4

= 48 square feet

Area of bottom rectangle shaded pink = (24 x 8) - 21 - 48

= 192 - 69

= 123 square feet

Material cost = $1.50 per square feet

Required cost = (123 + 72) x 1.50

= 195 x 1.50

= $292.5

Problem 5 :

Which is larger: a square cake pan that measures 8 inches on each side, or a circular cake pan with a diameter of 9 inches?

Solution :

Radius of circular portion = 9/2

= 4.5 inches

Side length of square = 8 inches

Area of square = side length x side length

= 8 x 8

= 64 square feet

Area of circle = πr²

= 3.14 (4.5)²

= 3.14 (20.25)

= 63.585 square feet.

= 64 square feet.

Both square and circle will have same area.

Problem 6 :

The diagram below shows one exterior wall of a house. The wall has a door that measures 3 ft. by 7 feet, and three windows which each measure 4 ft. by 4 ft. The wall (but not the doors or windows) will be covered with siding material. How much siding material is required?

A. 192 sq. ft.   B. 155 sq. ft.   C. 123 sq. ft.    D. 261 sq. ft.

Solution :

Area of square window = side x side

= 4 x 4

= 16 square feet

Area of 3 window = 3(16)

= 48 square feet

Area of rectangular door = length x width 

= 3 x 7

= 21 square feet

Area of figure above = 24 x 8 - 21 - 48

= 192 - 21 - 48

= 192 - 69

= 123 square feet

Problem 7 :

The figure below shows the plan for the a new county park. How many acres will the park occupy? (Use the estimate: 1 acre = 40,000 square feet.)

A. 6.53 acres        B. 391950 acres

C. 176.38 acres     D. 19.60 acres

Solution :

Area of parallelogram = base x height

= 1005 x 780

= 783900

1 acre = 40,000 square feet

= 783900 / 40000

= 19.59 acres

So, the required area is approximately 19.60 acres.

Problem 8 :

Suppose that it costs $480 to build a rectangular wooden deck that measures 6 feet by 8 feet. Assuming that the cost of such an object depends upon its size (area), how much would it cost to build a similar deck measuring 18 feet by 24 feet?

A. $640     B. $1440     C. $1,920      D. $4,320

Solution :

It is useful, in a problem involving two similar geometric figures, to compare their sizes via division. In this case we will divide the size (area) of the larger rectangle by the size of the smaller rectangle.

Size of larger deck = LW

= (18 feet)(24 feet)

= 432 square feet.

Size of larger deck = LW

= (6 feet)(8 feet)

= 48 square feet.

= (area of larger deck)/(area of smaller deck)

= 432/48

= 9

This means that the larger deck is nine times as large as the smaller deck, so it will cost nine times as much,

Cost of larger deck = 9($480)

= $4320.

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