SOLUTION: Find the length of the diagonal from the upper right corner of a room to its lower left corner. The height of the room is half its length. The area of the floor is 192 square feet
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Question 57691: Find the length of the diagonal from the upper right corner of a room to its lower left corner. The height of the room is half its length. The area of the floor is 192 square feet and the length of one of the floor's diagonals is 20 feet.
PLEASE HELP MEE
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Find the length of the diagonal from the upper right corner of a room to its lower left corner. The height of the room is half its length. The area of the floor is 192 square feet and the length of one of the floor's diagonals is 20 feet.
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It would help if your draw a rough diagram of this. Try to draw it as a 3 dimensional perspective. Label the length of the floor as x and the height of the room as .5x. Label the diagonal across the floor 20'
:
They are asking for the diagonal which goes from one corner on the floor, to the opposite corner in the ceiling. Look at your diagram, and you can see that this diagonal is the hypotenuse of a right triangle formed by the floor diagonal (given as 20') and the height, which we assigned as .5x.
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We need to find x.
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The area of the floor is given as 192 sq ft so we have;
x * width = 192
width = 192/x>
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We have the two sides formed by the right triangle on the floor:
x^2 + (192/x)^2 = 20^2
x^2 + (36864/x^2) = 400
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Get rid of the x^2 in the denominator, mult eq by x^2 and you have:
x^4 + 36864 = 400x^2
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x^4 - 400x^2 + 36864 = 0; a quadratic eq, solve for x^2
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Factors to:
(x^2 - 144)(x^2 - 256) = 0
x^2 = +144
x = 12
and
x^2 = 256
x = 16 is the length
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The reason we choose 16 as the length is because the height was given as .5(x); an 8' ceiling would be normal for a room, moreso that a 6' ceiling
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To answer the original problem of the room diagonal (h). Find the hypotenuse of the triangle:
h^2 = 20^2 + 8^2
h = SqRt[400 + 64)
h = SqRt[464}
h = 21.54 ft from lower corner to the upper corner
:
A lot of steps here, hope you could follow this.
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