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m_a_m_a [10]
3 years ago
11

Urgent, It is a Calculus question and I’ll appreciate your help. Thanks

Mathematics
1 answer:
BaLLatris [955]3 years ago
6 0

Answer:

  4733

Step-by-step explanation:

Please refer to the attached diagram.

Point A can be assigned x-coordinate "p". Then its y-coordinate is 6p^2. The slope at that point is y'(p) = 12p.

Point B can be assigned x-coordinate "r". Then its y-coordinate is 6r^2. The slope at that point is y'(r) = 12r.

We want the slopes at those points to have a product of -1 (so the tangents are perpendicular). This means ...

  (12p)(12r) = -1

  r = -1/(144p)

The slope of line AB in the diagram is the ratio of the differences of y- and x-coordinates:

  slope AB = (ry -py)/(rx -px) = (6r^2 -6p^2)/(r -p) = 6(r+p) . . . . simplified

The slope of AB is also the tangent of the sum of these angles: the angle AC makes with the x-axis and angle CAB. The tangent of a sum of angles is given by ...

  tan(α+β) = (tan(α) +tan(β))/1 -tan(α)·tan(β))

__

Of course the slope of a line is equal to the tangent of the angle it makes with the x-axis. The tangent of angle CAB is 2 (because the aspect ratio of the rectangle is 2). This means we can write ...

  slope AB = ((slope AC) +2)/(1 -(slope AC)(2))

  6(p+r)=\dfrac{12p+2}{1-(12p)(2)}\\\\3(p+r)(1-24p)=6p+1\qquad\text{multiply by $1-24p$}\\\\3\left(p-\dfrac{1}{144p}\right)(1-24p)=6p+1\qquad\text{use the value for r}\\\\3(144p^2-1)(1-24p)=144p(6p+1)\qquad\text{multiply by 144p}\\\\ 3456 p^3+ 144 p^2+ 24 p+1 =0\qquad\text{put in standard form}\\\\144p^2(24p+1)+(24p+1)=0\qquad\text{factor by pairs}\\\\(144p^2+1)(24p+1)=0\qquad\text{finish factoring}\\\\p=-\dfrac{1}{24}\qquad\text{only real solution}\\\\r=\dfrac{-1}{144p}=\dfrac{1}{6}

So, now we can figure the coordinates of points A and B, and the distance between them. That distance is given by the Pythagorean theorem as ...

  d^2 = (6r^2 -6p^2)^2 +(r -p)^2

  d^2 = (6(1/6)^2 -6(-1/24)^2)^2 +(1/6 +1/24)^2 = 25/1024 +25/576 = 625/9216

Because of the aspect ratio of the rectangle, the area is 2/5 of this value, so we have ...

  Rectangle Area = (2/5)(625/9216) = 125/4608 = a/b

Then a+b = 125 +4608 = 4733.

_____

<em>Comment on the solution</em>

The point of intersection of the tangent lines is a fairly messy expression, and that propagates through any distance formulas used to find rectangle side lengths. This seemed much cleaner, though maybe not so obvious at first.

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If ON = 7x - 5, LM = 6x + 4, NM = x + 6, and OL = 8y + 3, find the values of X and Y for which LMNO must be a parallelogram. Ple
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The values of x and y that will make LMNO a parallelogram are: x = 9, y = 1.5.

<h3>Properties of a Parallelogram</h3>
  • The opposite sides of a parallelogram are parallel to each other and are also of equal lengths.
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Therefore:

ON = LM

Substitute

7x - 5 = 6x + 4

Add like terms

7x - 6x = 5 + 4

x = 9

OL = NM

Substitute

8y + 3 = x + 6

Plug in the value of x

8y + 3 = 9 + 6

8y + 3 = 15

8y = 15 - 3

8y = 12

y = 12/8

y = 1.5

Therefore, the values of x and y that will make LMNO a parallelogram are: x = 9, y = 1.5.

Learn more about parallelogram on:

brainly.com/question/7200842

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\sqrt{(4^2 + 1.143^2)} =4.160 miles

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Step-by-step explanation:

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