All categories

3 years ago

Can anyone answer this for me?

Mathematics

1 answer:

kramer3 years ago

7 0

Answer:

$\frac{3}{2}$ π (or 270°)

Step-by-step explanation:

I use the trick with the coordinates.

(0, -1)

cos270° = 0 (the x-coordinate)

sin270° = -1 (the y-coordinate)

This trick works depending on the radius.

You might be interested in

Question 5 of 5

Flura [38]

The slope and y-intercept of this function is

B. The slope is 5/6, The y-intercept is (0, -3

<h3>How to find the slope and the y-intercept</h3>

Linear function is a function of the form y = mx + c

m = slope

c = intercept

The slope is represented as m, The slope by definition is the ratio change of the output values to the input values

the slope, m calculated using the points (0, -3) and (6, 2)

m = (y₀ - y₁) / (x₀ - x₁)

m = (2 - -3) / (6 - 0)

m = (5) / (6)

m = 5/6

c = y intercept, from the table

c = -3

Learn more about slope of linear functions at:

brainly.com/question/29323505

#SPJ1

8 0

1 year ago

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

BaLLatris [955]

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.