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Chapter 6: Problem 45

Find the angle \(\theta\) (in radians and degrees) between the lines. \(0.05 x-0.03 y=0.21\) \(0.07 x+0.02 y=0.16\)

Short Answer

Expert verified
The angle should be calculated by first finding the slopes of the two lines and then using the formula for the angle between two lines with a given slope. Afterwards, convert the angle from radians to degrees if needed.

Step by step solution

01

Finding the slope of line 1

First, let's rewrite the 1st equation, \(0.05 x-0.03 y=0.21\), in the form y=mx+b. After rearranging, we obtain the equation \(y=\frac{5}{3}x - 7\). So, the slope of line 1 is \(m_1 = \frac{5}{3}\).
02

Finding the slope of line 2

Next, let's rewrite the 2nd equation, \(0.07 x+0.02 y=0.16\), in the form y=mx+b. After rearranging, we obtain the equation \(y=-\frac{7}{2}x +8\). So, the slope of line 2 is \(m_2 = -\frac{7}{2}\).
03

Finding the angle between the two lines

The formula to find the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is \(\theta = \text{atan} \left( \frac{m_2 - m_1}{1 + m_1 \cdot m_2} \right)\). Substituting the slopes into this formula, calculate the angle in radians.
04

Converting the angle to degrees

If required, convert the angle obtained in radians into degrees by using the conversion \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\).

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