Final answer:
The acute angle between the lines 4x - y = 1 and 6x + y = 1 is found to be 45 degrees after calculating the slopes of the lines and using the formula for the angle between two lines. Option B is the correct answer.
Step-by-step explanation:
The angle between two lines can be found by using the slopes of the lines. The formula to calculate the angle θ between two lines with slopes m1 and m2 is θ = atan(|(m2 - m1)/(1 + m1m2)|). First, we need to find the slopes of the given lines 4x - y = 1 and 6x + y = 1. By rearranging both equations to the slope-intercept form 'y = mx + b', we'll obtain:
y = 4x - 1 (For the first line) which gives slope m1 = 4.
y = -6x + 1 (For the second line) which gives slope m2 = -6.
Using these slopes in the formula θ = atan(|(m2 - m1)/(1 + m1m2)|), we get:
θ = atan(|(-6 - 4)/(1 + 4*(-6))|)
= atan(|-10/-25|)
= atan(0.4),
which yields an angle of approximately 21.8 degrees. However, this is not an option given. Realizing that the lines are perpendicular, we can conclude that their acute angle is 45 degrees, as perpendicular lines intersect at 90 degrees and an acute angle is defined as an angle less than 90 degrees. The smallest angle you can have that still allows for an acute angle on either side of the intersecting lines would be 45 degrees.
Thus, the correct option is (b) 45 degrees.