Question 209795
diagonal is {{{sqrt(6m)}}}
in a right triangle, hypotenuse squared = width squared plus length squared.
in this rectangle, width = h + 2
in this rectangle, length = height = h
the diagonal of the gate is the hypotenuse of a right triangle.
your formula is:
{{{w^2 + h^2 = d^2}}}
where:
w is the width of the gate
h is the height of the gate
d is the diagonal of the gate.
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since you know that d = {{{sqrt(6)}}}, then d^2 must be equal to 6 meters because {{{sqrt(6)^2 = 6}}}
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since you know that w = h+2, then you can substitute h+2 for w.
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your formula becomes:
{{{(h+2)^2 + h^2 = 6}}}
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since:
{{{(h+2)^2 = h^2 + 4h + 4}}}
your formula becomes:
{{{h^2 + 4h + 4 + h^2 = 6}}}
combine like terms to get:
{{{2h^2 + 4h + 4 = 6}}}
subtract 6 from both sides of the equation to get:
{{{2h^2 + 4h - 2 = 0}}}
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solve using the quadratic formula to get:
h = .414213562 meters.
this makes:
w = 2.414213562 meters.
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this checks out because {{{h^2 + w^2 = sqrt(6)^2}}}
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quadratic formula is:
{{{h = (-b +- sqrt(b^2-4ac))/(2a)}}}
the equation you are placing into this formula is:
{{{2h^2 + 4h - 2}}}
the factors you will be using are:
a = 2
b = 4
c = -2
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you replace the a in the formula with 2.
you replace the b in the formula with 4.
you replace the c in the formula with -2.
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then you solve.
you will get one positive answer and one negative answer.  the negative answer is no good because h = height can't be negative.  only the positive answer is good.  that value should be the value i calculated for you.
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