Answer: k = -8.66
Explanation:
We have the equation:
k*x + 5*y = 10
First, let's rewrite this as a line in the slope-intercept form:
5*y = 10 - k*x
y = 10/5 - (k/5)*x
y = -(k/5)*x + 2
Now, we want to make an angle of 30° with respect to the y-axis.
Then we can think on a triangle rectangle, where we know one angle of 30°, the adjacent cathetus will be equal to the difference between the y-component and the y-intercept, and the opposite cathetus will be equal to the x-component.
To get this, we first need to evaluate the function in one given point, let's use x = 5. (we could use any value of x here, i use 5 because it is easier)
y = -(k/5)*5 + 2
y = -k + 2
then the cathetus of our triangle will be:
adjacent cathetus: (y - 2) = (-k + 2 - 2) = -k
opposite cathetus: x = 5.
Now we know the relation:
Tan(a) = (opposite cathetus)/(adjacent cathetus)
where a = 30°
Then:
Tan(30°) = 5/(-k)
-k = 5/tan(0.577) = 8.66
k = -8.66
This is the value of k that we wanted to find.