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Question 193878:
Answer by jim_thompson5910(35256) About Me  (Show Source):
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# 1

Let
x = measure of < PQR and
y = measure of < ABC


Since these two angles are complementary angles, this means that the two angles add to 90 degrees. So x%2By=90. Let's label this equation 1.


Also, because "< PQR is nine times as large as < ABC", we know that x=9y. We'll call this equation 2.



x%2By=90 Start with the first equation


9y%2By=90 Plug in x=9y


10y=90 Combine like terms on the left side.


y=%2890%29%2F%2810%29 Divide both sides by 10 to isolate y.


y=9 Reduce.


x=9y Go back to the second equation


x=9%289%29 Plug in y=9


x=81 Multiply


So the solutions are x=81 and y=9 which means that the measures of < PQR and < ABC are 81 and 9 degrees respectively.


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# 2

Take note that the angle marked 68 degrees and angle 3 are vertical angles. So this means that these two angles are equal.

So angle 3 is also 68 degrees.



Now because lines l and m are parallel, this means that the alternate interior angles are equal. Since angles 3 and 5 are alternate interior angles, this tells us that

angle 3 = angle 5

So angle 5 is 68 degrees as well.


Now remember, the sum of the 3 angles of ANY triangle is ALWAYS 180 degrees. So if we let the measure of angle 6 be "x", this means that

40 + angle 5 + angle 6 = 180


Plug in the given values to get:


40 + 68 + x = 180


Combine like terms to get


108 + x = 180


Subtract 108 from both sides

x = 180 - 108


and combine like terms to get

x = 72


So the measure of angle 6 is 72 degrees.


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# 3
a)

First, let "x" be the length of AC to get the triangle:





To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is a%5E2%2Bb%5E2=c%5E2 where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are x and 12 this means that a=x and b=12


Also, since the hypotenuse is 13, this means that c=13.


a%5E2%2Bb%5E2=c%5E2 Start with the Pythagorean theorem.


x%5E2%2B12%5E2=13%5E2 Plug in a=x, b=12, c=13


x%5E2%2B144=13%5E2 Square 12 to get 144.


x%5E2%2B144=169 Square 13 to get 169.


x%5E2=169-144 Subtract 144 from both sides.


x%5E2=25 Combine like terms.


x=sqrt%2825%29 Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


x=5 Take the square root of 25 to get 5.


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Answer:


So the solution is x=5 which means that AC = 5.

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b)

Now that we know the three sides are of length 5, 12, and 13 units, simply add them up to get

Perimeter = 5 + 12 + 13 = 17 + 13 = 30

So the perimeter is 30 units.

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c)

Recall, the area of a triangle is A=%28bh%29%2F2


A=%28bh%29%2F2 Start with the given formula.


A=%2812%2A5%29%2F2 Plug in the given base and height.


A=60%2F2 Multiply.


A=30 Divide.


So the area is 30 square units.


Note: this is just a coincidence that the perimeter and the area are equal.



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# 4

First, we need to find the length of diagonal.


If we cut the garden in half along the diagonal, we'll have this triangle set up (where "x" is the length of the diagonal):





Remember, the Pythagorean Theorem is a%5E2%2Bb%5E2=c%5E2 where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are 18 and 24 this means that a=18 and b=24


Also, since the hypotenuse is x, this means that c=x.


a%5E2%2Bb%5E2=c%5E2 Start with the Pythagorean theorem.


18%5E2%2B24%5E2=x%5E2 Plug in a=18, b=24, c=x


324%2B24%5E2=x%5E2 Square 18 to get 324.


324%2B576=x%5E2 Square 24 to get 576.


900=x%5E2 Combine like terms.


x%5E2=900 Rearrange the equation.


x=sqrt%28900%29 Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


x=30 Take the square root of 900 to get 30.



So the diagonal is 30 meters long.



Now just multiply the length of the diagonal by the cost per linear meter to get:

30%2A12=360


So it will cost $360 to build the diagonal walkway.

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# 5

First, let's convert every given measurement to feet. Recall (or look up), the given equality:

1 yard = 3 feet

In other words, there are 3 feet in a yard


So to convert 16 yards to feet, simply multiply both sides of the equation given above by 16 to get

1*16 yards = 3*16 feet

16 yards = 48 feet


So there are 48 feet in 16 yards


Now go back to the original equation, but now multiply both sides by 70 to get:

1*70 yards = 3 * 70 feet

70 yards = 210 feet


So there are 210 feet in 70 yards


Note: to convert yards to feet, all we're doing is multiplying the given measurement in yards by 3


So the equivalent dimensions (in feet) are 48 feet by 210 feet

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Now let's find the area of the lawn:


The formula for the area of a rectangle is:

Area = Length * Width


which looks like: A=LW


A=LW Start with the given formula


A=48%2A210 Plug in the given dimensions (in feet)


A=10800 Multiply


So the area of the lawn is 10,800 square feet.


Now divide the area of the lawn by the area that ONE bag can cover to get:

10800%2F525=20.571


So it will take 21 bags to cover his lawn (I rounded to the nearest whole number since you can't buy half a bag)


Now multiply this figure by the cost per bag to get: 21%2A5.75=120.75


So it will cost him $120.75 to cover his lawn with fertilizer.

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# 6


Note: since the diameter is 22mm, we take half of this number to find the radius. So the radius is 11 mm (since 11 is half of 22)


Area:

Recall, the area of a circle is A=pi%2Ar%5E2


A=pi%2Ar%5E2 Start with the given formula.


A=pi%2A11%5E2 Plug in r=11 (see above)


A=121pi Square 11 to get 121. Also, rearrange the terms.


So the exact area is 121pi square milimeters.


Note: because pi is irrational, we cannot write out the exact value of pi. So we must leave it alone.

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Circumference:

Remember, the circumference of a circle is C=2%2Ar%2Api


C=2%2Ar%2Api Start with the given formula.


C=2%2A11%2Api Plug in r=11 (see above)


C=22%2Api Multiply 2 and 11 to get 22


So the exact circumference is 22pi milimeters.


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# 7

Note: since 10 mm = 1 cm, this means that 5 cm = 50 mm (just multiply both sides by 5)


Since the diameter is 50 mm, this means that the radius is 25 mm (the radius is half the diameter)


The formula for the volume of a cylinder is V=pi%2Ar%5E2%2Ah


V=pi%2Ar%5E2%2Ah Start with the given formula.


V=pi%2A25%5E2%2A140 Plug in r=25 and h=140


V=pi%2A625%2A140 Square 25 to get 625


V=87500pi Multiply and rearrange the terms.


V=274889.356 Replace pi with 3.14159265... and multiply. Note: this result is approximate


So the glass can hold about 274,889.356 cubic milimeters.


Now divide the result by 1000 to convert to convert to cubic centimeters to get:

274889.356%2F1000=274.889



Now round to the nearest tenth to get V=274.889

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Answer:

So the glass can hold about 274.9 cubic centimeters of water.


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# 8


Here's some terminology:

Vertex: A point representing a location (city, town, etc...). In general, what the vertex represents isn't important.

Edge: A line connecting two vertices (plural for vertex)

Odd Vertex: A vertex with an odd number of edges connecting to it

Even Vertex: A vertex with an even number of edges connecting to it



So here's one way to draw the graph:




Take note that...

Vertex A: even vertex with 2 edges connecting to it

Vertex B: odd vertex with 3 edges connecting to it

Vertex C: even vertex with 2 edges connecting to it

Vertex D: odd vertex with 3 edges connecting to it


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# 9

It is possible to travel through every door once (and only once) given that you either start in room F or end in room F (otherwise you can't do it).


So here are 6 possible scenarios (there are possibly more)

Scenario # 1:



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Scenario # 2:



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Scenario # 3:



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Scenario # 4:



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Scenario # 5:



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Scenario # 6:



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