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Of course :D

Since we know that the slope is 4, we can narrow done the possible side lengths. 

Also, we know that the line is 1 unit from the origin.

That will create a right triangle with a height of 4 and a base of 1. Therefore, the area is 4.

Hope this helped!

Thanks for the help!!!

Hint: 

They are isosceles triangles

And If you know the measure of one angle, what can you do?

You can also try to find the measures of the two other angles by doing 120/2 which would get you 60 degrees!

Hope this helped!

If you know you posted there, you can go to your profile and see the "answers" page.

This is what I got--

(6x^3*4)+(4x^2*3x)+(-7x*3x^2)+(-5*6x^3)
24+12-21-30 = -15

Therefore, the coefficient would be -15.

Hope this helped!

 

Wait I think I found it! Here it is, If I'm not mistaken:

https://web2.0calc.com/questions/find-the-3-smallest-positive-x-intercepts-of-the

You can see my answer to this question here:

https://web2.0calc.com/questions/can-t-solve-this-blame-skyrim

o.o I tried looking for cosine stuff on web2.0 to no avail! Sorry :/

I agree! I think there might be an easier way though....

thx so much!

This is true, as the domain of arctan can accept any real number. the range however is limited to [-pi/2, pi/2] so it is continuous and 1-1 as a function.

Hey guest!

Since we know that \(g(x) = x^3+5x^2+9x-2\), we already have part of the desired expression of f(g(x)).

The question then becomes reduced to: 

\(f(x^3+5x^2+9x-2)\)

since \(f(x) = x^3-6x^2+3x-4\), this may seem really complicated at first glance. However, remember what they ask us for; the constant term. That means we can ignore everything except for the constant term of the expression \(x^3+5x^2+9x-2\), which is -2.

Plugging -2 into the function \(f(x)\), we get:

\(f(-2) = (-2)^3-6(-2)^2+3(-2)-4 = -8-24-6-4 =-42\)

Thus, the constant term of the function f(g(x)) is -42

True, as the inverse sine allows any x value between -1 and 1(that's the max and minimum of sine). The range is also all real numbers, as sine can go negative or positive.

Do you know when this was from?

Thanks for replying other guest, but the triangle is not just a right triangle with base 1 and height 4. I thought that at first but sadly, that is too simple. I still need help.

If this is difficult for you to visualize mentally, then draw it. 

Draw four X-Y coordinate planes. 

Put a dot on each axis 1 unit from the origin.  The origin is where the axes cross. 

Through each dot, draw a line with slope 4.  Slope 4 means the line goes up 4 for every 1 it goes to the right. 

My post accounted for two of the triangles.  I should have drawn them, cuz I neglected to mention the other two. 

_.

No problem!

Hey guest!

Let's first start off with an observation:

\(9x^{2}+24xy + 16y^2 = (3x+4y)^2\)

As such,

Part A) Because it says he arranges the marble as a square array, it makes sense that the total is a square number(of which it is). Since we know that 

\(9x^{2}+24xy + 16y^2 = (3x+4y)^2 \), that's how many total marbles there are. To get the number of marbles on each side of the square array, we just take the square root of given number, which gives us:

\(\sqrt{(3x+4y)^2} = 3x+4y\) marbles on each side of the array. The key takeaway here is that since the total number of marbles is a square number, that means since in a square array each side has an equal amount of marbles, the number squared must equal the total. To get the number, we just square root the total to get how many are on each side.

Part B) Since we already have the equation \(3x+4y\) as the number of marbles on each side of the square array, this problem becomes trivialized: it then becomes a simple question of substitution, where we get: 

\(3*(2) + 4*(5) = 6 + 20 = 26\)marbles on each side of the array

thanks for your help!

Hi all. The question is asking to find the area of the triangle in this diagram. Ignore the decimal values, I just wrote those to help visualize. The right triangle formed by the two axes and the line. As you can see, It will not be a whole number. {https://www.desmos.com/calculator/xq3u4ubj6x} This link is my diagram. Again, it is just to help visualize.

Hey guest!

I'm not really sure what the question means when the line has a distance of 1 unit from the origin. Does that mean the y intercept or x intercept, or what exactly? If it means that the line stops at 1 unit from the origin, that would just be a vector, so I'm not really sure what the question is asking here. If you have any ideas, feel free to lmk!

Locate that line so that it crosses the X-axis at –1 .... the slope of +4 causes the line to cross the Y-axis at +4. 

That creates a right triangle with a base of 1 unit and a height of 4 units.  It should be easy to figure the area. 

It would work at X=+1 as well, it's just easier to visualize the triangle if you use the Y-axis as one leg.

_.

Hey there guest! First, let's start the problem with the knowledge of a (pretty common) theorem in geometry: The power of a point theorem. There are three variations to this theorem, but in this problem, we only need to look at 1 : the variation which involves the tangent of a circle and a secant(a line intersecting two points on the circle. 

Firstly, notice we can extend the line QS to intersect the other side of the circle at Z. This is where power of a point comes in handy:

By power of a point:

\(QP^{2} = QR*QZ\)

Substituting known values in:

\(6^{2} = 3 * QZ\)

Dividing by 3, we get:

\(12 = QZ\)

Then RZ = QZ - 3(since that's the length of QR),

RZ = 12 - 3 = 9

Now that we have the length of a chord of Circle O(RZ is a chord), we can use power of a point once again, except another of its variations: with two intersecting chords. 

A handy trick to realize is that the circle's diameter is also a chord, meaning we could find the length with power of a point theorem. 

If we draw a diameter intersecting S, we could make power of a point easier to work with. By power of a point:\((R+2)*(R-2) = RS*SZ = 3*6 = 18\)

where R is the radius of the circle

This simplifies to: 
\(R^{2}-4 = 18\)

\(R^{2} = 22 \)

\(R= \sqrt{22}\)