See below for how I got this answer.
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Place the person at the origin (0,0). Specifically place their hand (that holds the leash) at this location.
Let's have the person face directly north which we'll define as the positive y direction. The positive x direction will be directly east. Think of a navigational compass.
If we draw out the vectors, then we'll have this
Let,
u = green vector pulling to the right
v = blue vector pulling to the left
so vector u is the vector representing the dog pulling 25 degrees to the person's right, and vector v is the the vector representing the dog pulling 35 degrees to the person's left.
Now that we have the drawing in place, let's modify it a bit. I'm going to break things down to have each vector on its own grid like shown below.
As you can see above, vector u forms an angle of 65 degrees with the positive x axis (see figure 2). Note how 65+25 = 90. So the theta value for vector u is 65 degrees.
Similarly, in the drawing above, we see that vector v has theta = 125 degrees (see red angle in figure 3) because vector v forms this angle with the positive x axis. Note how 90+35 = 125.
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In summary so far, we have
vector u with r = 15, theta = 65
vector v with r = 20, theta = 125
So we can convert each vector to cartesian form (x,y). This will allow us to easily add up the vectors to get the resultant vector. From there we will find the length of this resultant vector to get our final answer.
For now let's convert to cartesian form (x,y)
We'll use these formulas
x = r*cos(theta)
y = r*sin(theta)
Let's do one vector at a time
Vector u: r=15 and theta=65
x = r*cos(theta)
x = 15*cos(65)
x = 6.3392739261105
y = r*sin(theta)
y = 15*sin(65)
y = 13.5946168055498
So the tip of vector u is at the point (x,y) = (6.3392739261105, 13.5946168055498) which is an approximation.
Let's denote this as
(ux, uy) = (6.3392739261105, 13.5946168055498)
where ux is the x coordinate of vector u and uy is the y coordinate of vector u.
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Repeat these steps but now for vector v. We have r=20, theta=125 plugged into the following:
x = r*cos(theta)
x = 20*cos(125)
x = -11.471528727021
y = r*sin(theta)
y = 20*sin(125)
y = 16.3830408857799
So the tip of vector v is at the point (x,y) = (-11.471528727021, 16.3830408857799) which is an approximation.
Let's denote this as
(vx, vy) = (-11.471528727021, 16.3830408857799)
where vx is the x coordinate of vector v and vy is the y coordinate of vector v.
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Now add up the components of vectors u and v
(ux, uy) + (vx, vy) = (ux+vx, uy+vy)
(ux, uy) + (vx, vy) = (6.3392739261105+(-11.471528727021), 13.5946168055498+16.3830408857799)
(ux, uy) + (vx, vy) = (-5.1322548009105, 29.9776576913297)
The resultant vector terminates at the point (-5.1322548009105, 29.9776576913297) which is an approximation.
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Finally, we find the length of the resultant vector
Recall that the length of any general vector (a,b) is sqrt(a^2+b^2)
In this case, a = -5.1322548009105 and b = 29.9776576913297, so,
length of resultant = sqrt( (-5.1322548009105)^2 + (29.9776576913297)^2 )
length of resultant = 30.4138126514912
This is the final answer. Round this however you need to. This is an approximation. The units of this answer is in pounds. This is the min amount of force needed to keep the dogs from moving. Any larger amount of force pulled on the leash, and the person will be pulling the dogs in the opposite direction of where the resultant vector is pointing.