A certain right triangle has area 30 in. squared . one leg of the triangle measures 1 in. less than the hypotenuse. let x repres
ent the length of the hypotenuse. a) express the length of the leg mentioned above in terms of x. give the domain of x. b) express the length of the other leg in terms of x. c) write an equation based on the information determined thus far. square both sides and then write the equation with one side as a polynomial with integer coefficients, in descending powers, and the other side equal to 0. divide out any common factors to find the most simplified form. d) solve the equation in part (c) graphically. find the lengths of the three sides of the triangle.
1 answer:
A) The length of the longer leg is x-1
b) Based on the area, the other leg is 2*30/(x -1). Based on the Pythagorean theorem, the other leg is √(x^2 -(x -1)^2).
c) Equating the two expressions for the shorter leg, we have
.. 60/(x -1) = √(2x -1)
.. 3600/(x -1)^2 = (2x -1)
.. (2x -1)(x^2 -2x +1) = 3600
.. 2x^3 -5x^2 +4x -3601 = 0
d) There is one positive real root, at x=13. A graphical solution works well.
The three sides of the triangle are 5 in, 12 in, 13 in.
_____
5-12-13 is a well-known Pythagorean triple. It is the next smallest one after 3-4-5.
b) Based on the area, the other leg is 2*30/(x -1). Based on the Pythagorean theorem, the other leg is √(x^2 -(x -1)^2).
c) Equating the two expressions for the shorter leg, we have
.. 60/(x -1) = √(2x -1)
.. 3600/(x -1)^2 = (2x -1)
.. (2x -1)(x^2 -2x +1) = 3600
.. 2x^3 -5x^2 +4x -3601 = 0
d) There is one positive real root, at x=13. A graphical solution works well.
The three sides of the triangle are 5 in, 12 in, 13 in.
_____
5-12-13 is a well-known Pythagorean triple. It is the next smallest one after 3-4-5.
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Answer:
a)
And for this case we can rewrite the model like this:
If we integrate both sides we got:
If we use exponentials for both sides we got:
For this case and r = -0.16
So then our model would be given by:
Where t represent the number of hours
b)
Step-by-step explanation:
Part a
For this case we can assume the proportional model given by:
And for this case we can rewrite the model like this:
If we integrate both sides we got:
If we use exponentials for both sides we got:
For this case and r = -0.16
So then our model would be given by:
Where t represent the number of hours
Part b
For this case we can replace the value t=5 into the model and we got: