A MANUFACTURING PLANT WOULD LIKE TO EXPAND BUT NEEDS MORE SE

A MANUFACTURING PLANT WOULD LIKE TO EXPAND BUT NEEDS MORE SEWER CAPACITY, THE CITY HAS AGREED TO HELP BY INSTALLING A NEW SEWER LINE ACROSS A RECTANGULAR SHAPED VACANT LOT TO A NEW MANHOLE FOR THE PLANT\'S CONSTRUCTION AND MAINTENANCE PURPOSES. THIS AREA IS TO BE, PLACED ON THE VACANT LOT SUCH THAT THE OPPOSITE SIDES PASS THROUGH THE DIAGONAL CORNERS OF THE LOT. THE OWNER OF THE VACANT LOT WOULD LIKE MORE INFORMATION BEFORE SELLING THE SEWER AREA TO THE CITY FIND: DISTANCE AC = DISTANCE AE = DISTANCE CE = AREA ABCE = AREA CDE =

Solution

Step - 1:

Distance AC = ?

lower portion of the given rectangle is a right angle triangle

with sides given as 389.9 and 430 units

side AC can be found from pythogorous theorem

AC = sqrt(389.9^2+430^2)

= 579.84

Step - 2:

To find distance AE. Let distance AE = x, so BC = x

from the upper right angle triangle (CDE) we can get side CE as

CE = sqrt (430^2+[389.9-x]^2)

and BA = CE (propertiey of parallelogram)

Area of given rectangle = Area of upper traingle (CDE) +area of lower triangle +area of the parallelogram

389.9*430= (1/2)*430*(389.9-x) + (1/2)*430*(389.9-x) + sqrt (430^2+[389.9-x]^2)*(79.73)

389.9*430= 430*(389.9-x) + sqrt (430^2+[389.9-x]^2)*(79.73)

389.9*430= 430*389.9-430*x + sqrt (430^2+[389.9-x]^2)*(79.73)

0= -430*x + sqrt (430^2+[389.9-x]^2)*(79.73)

430*x = sqrt (430^2+[389.9-x]^2)*(79.73)

430*x/79.73 = sqrt (430^2+[389.9-x]^2)

5.39x = sqrt (430^2+[389.9-x]^2)

squaring on both sides

29.086 x^2 = (430^2+[389.9-x]^2)

29.086 x^2 = 430^2 + 389.9^2 - 2*389.9 x +x^2

28.086 x^2 + 779.8x - 336922 = 0

find the roots and report the positive value of x

x = -124.28 and 96.52

so positive root is 96.52

Distance AE = 96.52

Step-3:

Distance CE = ?

CE = 389.9 - x = 389.9 - 96.52 = 293.38

Step - 4:

Area ABCE (Parallelogram) = base * height

here height means wide

Area = CE * 79.73

= 293.38*79.73 = 23391 square units