SOLUTION: Hello, I managed to find the solution to this question through trial and error, but I am very curious to see how it is properly solved. <br> This is from my Precalculus book pub

Question 95181This question is from textbook Precalculus Mathematics for Calculus
: Hello, I managed to find the solution to this question through trial and error, but I am very curious to see how it is properly solved.

This is from my Precalculus book published by brooks/cole. (Precalculus Mathematics for Calculus Fifth Edition by Stewart Redlin and Watson)

" 78. Dimensions of a Lot
A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot? "

After hours of attempts and searching for clues I found a web site that let me plot the two sides and gave me the length of the hypotenuse. I finally came up with this: one side 49 ft, the other side 168 ft, and the hypotenuse 175 ft. 175+168+49=392 175-168=7

Thanks in advance
-Adam This question is from textbook Precalculus Mathematics for Calculus

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Without going into all the detail, here's a way you can do it.
.
First let x be one leg of the triangle. Then the hypotenuse is x + 7. Call the other leg
of the triangle y.
.
This means that the perimeter of the triangle (which is given as 392 ft) is the sum of the
three sides. In equation form this is:
.

.
Combine the x terms on the left side to get
.

.
Solve for y by subtracting 2x + 7 from both sides to get:
.

.
Now let's switch from the perimeter to the Pythagorean theorem. The sum of the squares of
the two legs equals the square of the hypotenuse. In equation form, for this problem the
Pythagorean theorem can be expressed in equation form as:
.

.
Square out the right side and you get:
.

.
Subtract from both sides and you have:
.

.
Solve for y by taking the square root of both sides:
.

.
So now we have two equations for y:
.
and

.
Since the left sides of these two equations are equal (both left sides are y), the right
sides must also be equal. So set the right sides equal:
.

.
Get rid of the radical by squaring both sides to get:
.

.
Subtract 14x + 49 from both sides:
.

.
Rearrange terms in descending powers of x:
.

.
This is now in the standard form of a quadratic equation. Use the quadratic formula to
solve it and you will get two answers for x. The two answers are:
.
and

.
The first answer gets tossed out because if x = 220.5 then the hypotenuse is 227.5 and
adding just these two gives an answer that is bigger than the perimeter.
.
So the acceptable answer is (as you already knew) x = 168. This makes the hypotenuse
175, and therefore, y = 49.
.
Hope this helps.
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