2. Are the expressions -5 + -2 and -5 – 2 equivalent? Why or why not? Explain and show your work.
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Answer:
Given: In triangle XYZ, , XY=13x-21 , YZ=8x-6 and XZ=x+4.
If two angles are congruent if and only if, they measure the same number of degrees.
therefore, from the given condition; .
Isosceles Triangle : An triangle with two equal sides. The angles opposite the equal sides are also equal.
Therefore, the triangle XYZ is an isosceles triangle, as the base angle and
are equal and the sides opposite to these angles are also equal i.e, XY=YZ.
Since, we have XY=YZ ,
⇒ 13x-21 =8x-6 or
13x-8x=21-6 or
5x=15
Simplify:
x=3
Therefore, the sides of an isosceles triangle are:
XY = 13x-21 = = 39-21 = 18 unit ,
YZ = 8x-6 = = 24-6 =18 unit,
and XZ = x+4 = 3+4 =7 unit.
Since, the triangle is isosceles, we draw a line from an vertex angle Y of a triangle XYZ which is perpendicular (i.e, of 90 degree angle) to opposite side meets the opposite side at its midpoint i.e, say W.
As W is the midpoint(i.e, it divide the sides into two equal halves),
XW=WZ = unit
Now, use the Pythagorean theorem, to find the altitude WY.
⇒
Substitute the value of WZ = 3.5 unit and YZ = 18 unit in above formula to calculate the length of WY;
⇒
Simplify:
or
On simplify we get;
WY=17.65 unit(approx.)
The vertex angle Y is split into two equal angles, we can find the vertex angle by finding the one of the base angle by using the fact; .
⇒
⇒ (approx) or
Therefore, the angle .
The sum of the measure of the angle in the triangle is 180 degree.
In triangle XYZ.
Since
then,
⇒
Simplify:
.
Therefore, the measure of each angle are: and
We are given inequalities : 5x+y=>10
x+y<=6
x+4y=>12
x=>0
y=>0 .
Let us graph it first and the find the coordinate of the vertices of feasible region.
Coordinates of the feasible region are (1,5) , (1.474, 2.632) and (4,2).
Now, we need to plug those cordinates (1,5) , (1.474, 2.632) and (4,2) in the given function c=10,000x + 20,000y.
Let us plug those points one by one.
For (1,5)
c=10,000x + 20,000y = 10,000(1)+ 20,000(5)= 10,000+ 100,000 = 110,000.
For (1.474, 2.632)
C = 10,000(1.474)+ 20,000(2.632) = 67,380.
For (4,2)
C = 10,000(4)+ 20,000(2) = 80,000.
We got the minimum value 67,380 for (1.474, 2.632) coordinate.
Therefore, minimum for given function C=10,000x + 20,000y is 67,380.
