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xz_007 [3.2K]
3 years ago
5

Let us have four distinct collinear points $a,$ $b,$ $c,$ and $d$ on the cartesian plane. the point $c$ is such that $\dfrac{ab}

{cb} = \dfrac{1}{2}$ and the point $d$ is such that $\dfrac{da}{ba} = 3$ and $\dfrac{db}{ba} = 2.$ if $c = (0, 4),$ $d = (4, 0),$ and $a = (x, y),$ what is the value of $2x + y$?

Mathematics
1 answer:
Deffense [45]3 years ago
5 0

Start with a line segment connecting two points, A and B. \dfrac{DA}{BA}=3 means DA is 3 times longer than BA. Clearly, D cannot fall between A and B because that would mean DA is shorter than BA. So there are two possible locations where D can be placed on the line relative to A and B.

But with \dfrac{DB}{BA}=2, or the fact that DB is 2 times longer than BA, we can rule out one of these positions; referring to the attachment, if we place D to the left of A, then DB would be 4 times longer than BA.

Finally, \dfrac{AB}{CB}=\dfrac12, so that CB is 2 times longer than AB. Again we have two possible locations for point C (it cannot fall between A and B), but one of them forces C to occupy the same point as D. However, A, B, C, D are distinct, so C must fall to the left of A.

Now let d be the length of AB. Then the length of CD in terms of d is 4d. We have the coordinates of C and D, and the distance between them is \sqrt{(4-0)^2+(0-4)^2}=4\sqrt2. So

4d=4\sqrt2\implies d=\sqrt2

The slope of the line through C and D is

\dfrac{0-4}{4-0}=-1

and so the equation of the line through these points is

y-4=-(x-0)\implies x+y=4

So the coordinates of A are (x,y)=(x,4-x). The distance between C and A is d=\sqrt2, so we have

\sqrt{(x-0)^2+(4-x-4)^2}=\sqrt{2x^2}=|x|\sqrt2=\sqrt2\implies|x|=1

Since A falls to the right of C (in the x,y plane, not just in the sketch), we know to take the positive value x=1. Then the y coordinate is y=4-1=3.

All this to say that A is the point (1, 3), so

2x+y=2+3=5

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</span>Factor the expression.<span><span>(<span>h<span><span>−2</span>k</span></span>)</span><span>(<span><span>h+<span>2k</span></span>+4</span>)</span></span>

8 0
3 years ago
Set x is the set of all multiples of 6 between 15 and 70 what set is equal to set x
Pie

Answer:

<em>X ∈ { 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 206, 312, 318, 324, 330, 336, 342, 348, 354, 360, 366, 372, 378, 384, 390, 396, 402, 408, 414 }</em>

Step-by-step explanation:

See procedure below;

Set X { all multiples of 6 between 15 and 70, provided multiples belong to the set of natural numbers },

Set X { 6n | 15 < n < 70, n ∈ natural numbers },

n ∈ { 16, 17, 18, 19, 20, 21, 22, 23, 24, 25.........69 },

X ∈ { 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156.........414 }

<em>Answer; { 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 206, 312, 318, 324, 330, 336, 342, 348, 354, 360, 366, 372, 378, 384, 390, 396, 402, 408, 414 }</em>

8 0
2 years ago
Pablo wants to get a grade of at least 95 in math this semesterSo far he has test scores of 89 98, 97 and 92. He will take his l
Alex Ar [27]

Answer:

the answer is 97 if you're rounding but 99 if you want exactly 95

Step-by-step explanation:

if you add all the previous grades up it'll add up to 376 and then when you add 97 and then divide it by 5 (the total amount of assignments) it'll add up to 94.6 which rounds up to a 95 unless you want it a 95 exactly which would probably be 99

3 0
3 years ago
Read 2 more answers
Suppose a company produces report indicates that the surveyed distance between two points is 1200 feet with a margin of error of
Otrada [13]
<h2>Hello!</h2>

The answers are:

MaximumDistance=1200.1ft\\MinimumDistance=1199.9ft

<h2>Why?</h2>

Since we are given the margin of error and it's equal to ±0.1 feet, and we know the surveyed distance, we can calculate the maximum and minimum distance. We must remember that margin of errors usually involves and maximum and minimum margin of a measure, and it means that the real measure will not be greater or less than the values located at the margins.

We know that the surveyed distance is 1200 feet with a margin of error of ±0.1 feet, so, we can calculate the maximum and minimum distances that the reader could assume in the following way:

MaximumDistance=ActualDistance+0.1feet\\\\MaximumDistance=1200feet+0.1feet=1200.1feet

MinimumDistance=ActualDistance-0.1feet\\\\MaximumDistance=1200feet-0.1feet=1199.9feet

Have a nice day!

4 0
2 years ago
Can someone help me!!!!!!
Sholpan [36]
It's A since tangent is just opposite over adjacent, 12 is the opposite and 5 is the adjacent
7 0
2 years ago
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