Part 3 Module 8Real-world problems involving area

Useful facts

The following facts will be provided on a formula sheet when you take quizzes or tests on this material in lab.

LWArea = LWbhArea = bhbhArea = (1/2)bhrArea = πr2

Exercise #1

www.math.fsu.edu/~wooland/Geo56/Geo14.html

A rectangular section of wall measuring 22 feet by 8 feet will be covered with isosceles right triangular tiles with legs measuring 5 inches.Approximately how many tiles are needed to cover the section of wall?

A. 7B. 1014C. 169D. 2028

Solution #1

A rectangular section of wall measuring 22 feet by 8 feet will be covered with isosceles right triangular tiles with legs measuring 5 inches.Approximately how many tiles are needed to cover the section of wall?To answer a question of the form “How many of these smaller objects are needed to ‘equal’ this larger object, we divide the size of the larger object by the size of the smaller object.For a two-dimensional object, like a rectangle or triangle, “size” means “area.”The area of a rectangle is LW. The area of a triangle is (1/2)bh.For the rectangular floor, L = 22 ft and W = 8 ft.For the isosceles right triangular tiles, b = 5 inches and h = 5 inches.Before we calculate the areas, we must reconcile the discrepancy in units. If we use the units as currently listed, the area of the rectangle will be in square feet, but the area of the triangle will be in square inches, so the two areas won’t be comparable.

Solution #1, page 2

The area of a rectangle is LW. The area of a triangle is (1/2)bh.For the rectangular floor, L = 22 ft and W = 8 ft.For the isosceles right triangular tiles, b = 5 inches and h = 5 inches.We will convert the measurements for the rectangle from feet to inches before we calculate the area.L = 22 ft = 22 x 12 inches = 264 inchesW = 8 ft = 8 x 12 inches = 96 inches

Now, calculate the area of the rectangle and the area of the triangle.Area of rectangle = (264 inches)(96 inches) = 25,344 square inchesArea of triangle = (1/2)(5 inches)(5 inches) = 12.5 square inches(Area of rectangle)/(Area of triangle)= 25344/12.5= 2027.5It will take roughly 2028 tiles to cover the wall, because the wall is 2028 times as big as one tile.

Exercise #2

www.math.fsu.edu/~wooland/Geo56/Geo19.html

The figure below shows the plan for the new parking lot a Southwestdale Mall. It is estimated that such construction will cost $9.50 per square yard. Find the total cost.A. $4360.50B. $164888.33C. $1483995D. $494665

Solution #2

To answer this question, we need to find the area of the parallelogram, and then multiply the area by $9.50.

We want the area to be in square yards.

This means that we will convert the two measurements (b = 615 feet, h = 762 feet) from feet to yards before we use the formula A=bh.

Convert measurements from feet to yards:b = 615 feet = 615/3 yards = 205 yardsh = 762 feet = 762/3 yards = 254 yards

Area = (205 yards)(254 yards) = 52,070 square yardsCost = 52,070 square yards x $9.50 per square yards = $494,665

Exercise #3

Find the area contained within the race track shown below.

A. 97,489 square yardsB. 296,378 square yardsC. 135,412 square yardsD. 45,137 square yardsE. 32.498 square yards

782 feet374 feet

Solution #3

Because the want to calculate the area of this figure in square yards, but the measurements are in feet, we will convert the two measurements from feet to yards, first.782 feet = 782/3 yards = 260.67 yards374 feet = 374/3 yards = 124.67 yardsWe will assume that the figure consists of a rectangle with a half-circle joined to each end. Thus,Area of figure = area of rectangle + areas of two half-circles= area of rectangle + area of one complete circleArea = LW + πr2

From the figure, we see that L = 260.67 yards, W = 124.67 yards, r = 62.33 yards (radius is half of the diameter, and the diameter is 124.67 yards).A = (260.67)(124.67) + π(62.33)2 = 44,702.91 square yardsOur answer doesn’t exactly match any of the multiple-choice options, because we rounded some numbers early in the calculation. The correct choice is D. 45,137 square yards

260.67 yards124.67 yards

Exercise #4

Suppose that a tank truck hauling a liquid toxic substance springs a leak, leaving in the parking lot a circular puddle with a radius of 6 feet. It takes 36 hours for the specialists to decontaminate this spill. How long would it take to clean up a similar spill with a radius of 2 feet?

A. 6 hoursB. 12 hoursC. 4 hoursD. 9 hoursE. 18 hours

Solution #4

2 6

It takes 36 hours to decontaminate a circular puddle

of toxic waste with a radius of 6 feet.

How long will it take to decontaminate a similar spill

with a radius of 2 feet?

The answer IS NOT 12 hours , because the larger

circle is not three times as big as the small

circle (as the scale drawing below suggests).

There are several ways to arrive at the one correct answer. Any correct solution must take into account the area of the large circle, the area of the small circle, and the fact that the large circle requires 36 hours.We will take the same approach that we did with Exercise #1: compare the areas of the two circles, by dividing:(Area of large circle)/(area of small circle)= (π62 square feet)/(π22 square feet)= (36π square feet)/(4π square feet)= 9 (we have simplified the fraction by canceling common factors and units).

This means that the large circle is 9 times as big as the small circle, so it takes 9 times as long to decontaminate.

The small circle takes 1/9 as much time as the large circle.1/9 of 36 hours = 4 hours. The correct answer is C.

Exercise #5

www.math.fsu.edu/~wooland/Geo56/Geo23.html

The figure below shows the parcel of land on which Gomer the rancher confines his weasels. His rule of thumb dictates that each weasel requires 25 square meters of space. Approximately how many weasels can the parcel accommodate?

A. 6516050B. 733281C. 1173D. 28

95 m190 m190 m47.5 m

Solution #5Each weasel requires 25 square meters of space. Approximately how many weasels can the parcel accommodate?We need to find the area of the figure, in square meters, and then divide by 25.To find the area of the figure, we need to visualize the figure in terms of simpler figures, such as triangles and rectangles. One way to do this (not the only correct way) is to recognize that the figure is formed by cutting a right triangle off of the upper right-hand corner of a square.

95 m190 m190 m47.5 m190 m95 m190 m142.5 mArea of figure = Area of square – Area of triangle= 190 x 190 – [(1/2) x 95 x 142.5]= 36100 – 6768.75= 29331.25 square meters

Thus, the number of weasels is29331.25/25 = 1173.25The best choice is C. (1173)

Exercise #6

www.math.fsu.edu/~wooland/Geo56/Geo22.html

The figure below shows the parcel of land on which Gomer the rancher confines his weasels. He is going to enclose the parcel with a fence. Find the total cost, assuming that the fence will cost $3.00 per linear meter.

95 m190 m190 m47.5 mA. $1568B. $1743C. $2081D. $87994E. $65634

Solution #6He is going to enclose the parcel with a fence. Find the total cost, assuming that the fence will cost $3.00 per linear meter.In this case, we don’t need to find the area of the figure. This is a problem involving distance. We need to find the perimeter ( P ) of the figure, and then multiply by $3.00.

95 m190 m47.5 m190 mx P = 95 + 190 + 190 + 47.5 + x= 522.5 + x

To find x, we recognize that x is the hypotenuse of an “invisible” right triangle, whose legs measure 95 meters and 142.5 meters, respectively (see next slide).

Solution #6, page 2

P = 95 + 190 + 190 + 47.5 + x= 522.5 + xwhere x is the hypotenuse of an “invisible” right triangle, whose legs measure 95 meters and 142.5 meters, respectively, so we find x by using the Pythagorean Theorem.

95 m190 m190 m47.5 m95 m142.5 mx

€

x2=952+142.52x2=9025+20306.5x2=29331.25x=29331.25≈171.26P = 95 + 190 + 190 + 47.5 + x= 522.5 + 171.26 = 693.76 metersCost of fence = 693.76 meters x $3.00 per meter = $2081.