Find the slope of the power line.

The equation of the power line is $3x + 4y - 24 = 0$.
Rearrange to slope-intercept form $y = mx + c$:

The slope of the power line, $m_1$, is $-\frac{3}{4}$.

For perpendicular lines, the product of their slopes is $-1$.

The slope of the perpendicular path, $m_2$, is $\frac{4}{3}$.

The path passes through the workshop at $(2,2)$ and has a slope of $\frac{4}{3}$. Use the point-slope form $y - y_1 = m(x - x_1)$:

Multiply by 3 to clear the fraction:

Rearrange to the general form $Ax + By + C = 0$:

The equation of the perpendicular path is $\boxed{\mathbf{4x - 3y - 2 = 0}}$.

Equation of power line: $3x + 4y = 24$ (1)
Equation of perpendicular path: $4x - 3y = 2$ (2)

Multiply equation (1) by 3 and equation (2) by 4 to eliminate $y$:

Add the two new equations:

Using equation (2):

The coordinates of the intersection point are $\boxed{\mathbf{\left(\frac{16}{5}, \frac{18}{5}\right)}}$ or $\boxed{\mathbf{(3.2, 3.6)}}$.

The workshop is at $(x_0, y_0) = (2,2)$.
The power line equation is $3x + 4y - 24 = 0$, so $A=3$, $B=4$, $C=-24$.

The formula is $D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.

The shortest distance from the workshop to the power line is $\boxed{\mathbf{2 \text{ units}}}$.

Shortest distance = 2 units.
Cost per unit = 1000.

Total cost = Shortest distance $\times$ Cost per unit
Total cost = $2 \times 1000$
Total cost = 2000
The total cost to reach the power line is $\boxed{\mathbf{2000}}$.