Question 175943: An airport, a factory, and a shopping center are the vertices of a right triangle formed by three highways.The airport and factory are 6 miles apart.Their distances from the shopping center are 3.6 miles and 4.8 miles respectively.A service road will be constructed from shopping center to the highway that connects the airport and factory.What is the shortest possible length for the service road?Round to the nearest hundredth.
Answer by gonzo(654)   (Show Source):
You can 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.
---
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.
---
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.
---
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.
---
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.
---
how do we find the length of this?
---
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.
---
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
---
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
---
either one will work.
---
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.
---
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
---
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
---
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
---
now that we have y, we can solve for x.
---
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
---
we now have:
y = 3.84
x = 2.88
---
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.
---
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:
---
the shortest possible length for the service road is 2.88 miles.
---
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.
---
|
|
|
