Question 101529
This can be solved using the Pythagorean theorem which states that in a right triangle
having legs A and B and hypotenuse C, then:
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{{{A^2 + B^2 = C^2}}}
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In this problem, the two legs of the right triangle are the width and the length of the 
rectangle, and the hypotenuse is the diagonal.
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The problem tells you that the diagonal is 4 cm. Substitute that value in for C, the hypotenuse,
and the equation becomes:
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{{{A^2 + B^2 = 4^2}}}
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The problem also tells you that if W represents the width of the rectangle, then W+1 is
the length because the length is 1 cm more than the width. So substitute W and W+1 into
the Pythagorean theorem and you then have:
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{{{W^2 + (W+1)^2 = 4^2}}}
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On the left side if you square out {{{(W+1)^2}}} you get {{{W^2 + 2W + 1}}}. Substitute 
this and the equation then becomes:
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{{{W^2 + W^2 + 2W + 1 = 4^2}}}
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On the left side, combine the two {{{W^2}}} terms and the equation is then:
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{{{2W^2 + 2W + 1 = 4^2}}}
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Square the right side:
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{{{2W^2 + 2W + 1 = 16}}}
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Get rid of the 16 on the right side by subtracting 16 from both sides to get:
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{{{2W^2 + 2W - 15 = 0}}}
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Notice that this is in the standard quadratic form of:
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{{{ax^2 + bx + c = 0}}}
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and, by comparing our equation and the standard form you can see that a = 2, b = 2, c = -15,
and W = x.
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You can use the quadratic formula to solve this equation. The quadratic formula says that
for equations of the form: {{{ax^2 + bx + c = 0}}} the answers will be given by:
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{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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Now all we have to do is to solve our equation is to replace x with W and substitute
the values we got for a, b, and c into the answer form. When you do that you get:
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{{{W = (-(2) +- sqrt( (2)^2-4*2*(-15) ))/(2*2) }}}
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First, let's simplify the terms under the radical. {{{2^2 = 4}}} and {{{-4*2*(-15) = +120}}}.
Substitute these results and you get:
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{{{W = (-(2) +- sqrt(4+120))/(2*2) }}}
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The two terms under the radical combine to 124 and the denominator of (2*2 = 4}}}. Substituting
these results into the equation gives:
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{{{W = (-(2) +- sqrt(124))/4 }}}
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Calculator time. The square root of 124 is 11.13552873
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{{{W = (-2 +- 11.13552873)/4}}}
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The answer for W must be positive because it makes no sense to have a negative width. So
we can drop the negative sign in front of the 11.13552873 to get:
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{{{W = (-2 + 11.13552873)/4}}}
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Combine the two numbers in the numerator:
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{{{W = 9.13552873/4}}}
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Calculator time. Divide out the fraction on the right side to get:
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{{{W = 2.28388218}}}
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which rounds off to the answer you got for the width of 2.284 cm
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Then all you have to do to find the length is add 1 cm and you get that the length is 3.284 cm.
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Hope this helps you get it down on paper.
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