Question 1165281: A vertical wall was built around a rectangular court to a certain height. The width of the adjacent side was 5 m. less than 6 times the height of the wall and the length of one side of the court was 2 m. less than 8 times the height of the wall. The floor area of the court was greater than the area of the inside face of the wall by 178m.
a.) Determine the length of the rectangular court.
b.) Determine the width of the rectangular court.
c.) Determine the number of gallons needed to paint the inside portion of the wall only. Use 4 coatings and assume one gallon has a spreading capacity of 400 sq.ft of wall.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the vertical wall is built to a certain height.
we'll call it x until we can find out what the value of x is.
you are given that the width of the court is equal to 5 meters less than 6 times the height of the wall.
the width of the court is therefore equal to 6x - 5.
you are given that the length of the court is equal to 2 meters less than 8 times the height of the wall.
the length of the court is therefore equal to 8x - 2.
the floor area of the court is equal to the length of the court times the width of the court.
since the length of the court is equal to 8x - 2 and the width is equal to 6x - 5, then the area of the the court is equal to (8x - 2) * (6x - 5).
this is equal to 48x^2 - 40x - 12x + 10
combine like terms to get the area of the court is equal to 48x^2 - 52x + 10.
the area of the wall on the length side of the court is equal to x * (8x - 2) = 8x^2 - 2x.
the area of the wall on the width side of the court is equal to x * (6x - 5) = 6x^2 - 5x.
the total area of the inside face of the wall is equal to 2 * (8x^2 - 2x) + 2 * (6x^2 - 5x) = 16x^2 - 4x + 12x^2 - 10x = 28x^2 - 14x.
you are given that the area of the court is 178 meters greater than the area of the inside face of the wall.
this means that 48x^2 - 52x + 10 = 28x^2 - 14x + 178.
subtract the right side of this equation from both sides of this equation to get:
48x^2 - 52x + 10 - 28x^2 + 14x - 178 = 0
combine like terms to get:
20x^2 - 28x - 168 = 0
factor this quadratic equation to get:
x = -2.1 or x = 4.
x can't be negative, so x = 4.
that's the height of the wall.
the area of the inside face of the wall is 28x^2 - 14x = 392 square feet when x = 4.
the area of the court is 48x^2 - 52x + 10 = 570 square feet when x = 4.
570 - 392 = 178.
the area of the court is 178 meters less than the area of the inside fact of he walls, as was given.
4 coats of paint on the walls requires 4 * 392 = 1568 square feet of wall to be painted.
1568 square feet divided by 400 square feet = 3.92 gallons of paint.
you would need 4 gallons of paint to cover the walls with 4 coats of paint and you would have a small amount of paint left over.
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