Question

Give numerical values for order-of-magnitude estimates for the following quantities. Explain and justify the reasonableness of the assumptions and approximations that you need to make.

(a) The number of cars that pass through an intersection of two busy streets during the evening commute on a typical workday

(b) The number of bricks that form the exterior of a large building on a university campus

(c) The volume of concrete in the sidewalks on a university campus

Answer 1

(a) We estimate the people moving in the evening commute as 20000 people near offices to the intersection. We then assume 1 car for every 4 people hence we calculate that around 5000 cars cross the intersection. Taking into considerations the uncertainties we estimate around 500 – 50000 cars pass through the intersection

 

(b) Consider the building to be square in shape with dimensions of 40 ft × 50 ft and a height of 60 ft. The building has four walls and assumes each wall is of thickness 2.25 in. Next we assume that the brick shape is 8 × 4 × 2.25 in. as the average size of bricks in the United States is 8 × 4 × 2.25 in. Hence we can approximate the number of bricks in each wall as

 

Area of each wall = 50 × 60

                             = 3000 ft2

Area of each brick = 8 in × 4 in

                               = 32 in2

 

Now convert the area of bricks in ft2

(32 in2) × (1ft2/144 in2) = 0.22 ft2

 

Therefore the number of bricks in each wall is estimated as given below:

Number of bricks = Area of wall/Area of brick

                             = 3000/0.22

                             = 13636 bricks

 

Hence each wall has 13636 bricks. 

Therefore the number of bricks in the exterior of the building is 13636 × 4 = 54544 bricks. Hence considering the uncertainties in the assumptions for the size of the bricks and the size of the building we take an order of magnitude of factor 5 and estimate that the number of bricks is within the range of 11000 – 200000 bricks.

 

(c) We first assume that the sidewalks cover the perimeter of the university which is in the shape of a square of size 100 × 100 ft. Hence there are four sidewalks corresponding to the four sides of the square. Each side walk is of size 100 × 2 ft and thickness of 1 ft.

 

Now the amount of concrete required is equal to volume of the sidewalks. Hence total volume of sidewalks is calculated as below

 

Volume of sidewalks = 4 × (100 × 2 × 1)ft3

                                   = 800 ft3

 

Hence volume of concrete in the sidewalks is 800 ft3

 

Therefore taking the uncertainties in the size of the sidewalks and the number of sidewalks the volume of concrete in sidewalks is in the range of 160 ft3 – 4000 ft3.

Therefore taking the uncertainties in the size of the sidewalks and the number of sidewalks the volume of concrete in sidewalks is in the range of 160 ft3 – 4000 ft3.