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put this solution on YOUR website!your triangle is AFS if you start at the top and go around clockwise from A to F to S, where A is the location of the airport, F is the location of the factory, S is the location of the shopping center.
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the shopping center is the right angle.
the distance from A to F is the hypotenuse.
the distance from S to A is one of the legs.
the distance from S to F is the other leg.
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if this is a right triangle as stated above, then (AS)^2 + (SF)^2 = (AF)^2.
(4.8)^2 + (3.6)^2 = (6)^2
if you check out the math using your calculator, you will find that this equation is true, so we have the right diagram.
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the shortest possible distance from the shopping center to the highway that connects the airport to the factory will be a line from point S to intersect with the line AF at a right angle.
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if you call the intersection point B, then we are talking about the line SB intersecting the line AF at a right angle at point B.
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how do we find the length of this?
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let x = the length of the line SB.
let y = the length of the line AB.
then:
6 - y = the length of the line BF.
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you have 2 equations in 2 unknowns that you have to solve simultaneously.
the first equation is:
x^2 + y^2 = 4.8^2
the second equation is:
x^2 + (6-y)^2 = 3.6^2
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you can substitute:
4.8^2 - y^2 for x^2
or you can substitute:
4.8^2 - x^2 for y^2
both these are derived from the equation:
x^2 + y^2 = 4.8^2
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either one will work.
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i substituted 4.8^2 - y^2 for x^2 and solved for y because i thought it would be easier to solve that way.
i could have been wrong, but it doesn't matter since i got the answer anyway.
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here's how it worked out.
the first equation was:
x^2 + y^2 = 4.8^2
subtract x^2 from both sides to get:
x^2 = 4.8^2 - y^2
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the second equation was:
x^2 + (6-y)^2 = 3.6^2
substituting 4.8^2 - y^2 for x^2, gets:
4.8^2 - y^2 + (6-y)^2 = 3.6^2
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multiplying out all terms within the parentheses gets:
4.8^2 - y^2 + 36 - 12*y + y^2 = 3.6^2
combining like terms gets:
4.8^2 + 36 - 12*y = 3.6^2
adding 12*y to both sides of the equation and subtracting 3.6^2 from both sides of the equation gets:
4.8^2 + 36 - 3.6^2 = 12*y
simplifying gets:
23.04 + 36 - 12.96 = 12*y
combining like terms gets:
46.08 = 12*y
dividing both sides by 12 gets:
y = 3.84
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now that we have y, we can solve for x.
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the first equation is:
x^2 + y^2 = 4.8^2
which becomes:
x^2 + 3.84^2 = 4.8^2
subtracting 3.84^2 from both sides and simplifying and combining gets:
x^2 = 4.8^2 - 3.84^2 = 23.04 - 14.7456 = 8.2944
x = square root of (8.2944) = 2.88
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we now have:
y = 3.84
x = 2.88
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x is good for the first equation because we used that equation to solve for x.
to make sure x is good for the second equation, we use the values of y and x in the second equation to see if it is true or false.
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the second equation is:
x^2 + (6-y)^2 = 3.6^2
this becomes:
2.88^2 + 2.16^2 = 3.6^2
this becomes:
8.2944 + 4.6656 = 12.96
which becomes:
12.96 = 12.96
since the equation is true, the value we found for x and y are good and the answer is:
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the shortest possible length for the service road is 2.88 miles.
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is there a shorter distance from the shopping center to the highway between the airport and the factory?
answer is no, because:
take any other intersection point on the line AF.
call it C.
the line SC will form a right triangle with the perpendicular line SB. the line SC will be the hypotenuse of that right triangle.
the hypotenuse of a right triangle is always greater than either leg.
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