I am new to this community, so feel free to point me in a different area if this is not appropriate here. It is a specific question, but I am looking for a general answer - i.e., formulae that I can use when the baseline distance, angles, and areas change.
I have a pasture that I want to split into two fields as shown in the sketch which I hope is linked. The width at the bottom is 764.81 feet, and the sides angle inwards slightly as shown. The 1 acre field on the left is 8' deeper than the 3 acre field beside it, and also excludes the 8' triangle. (1 acre is 43,560 square feet).
The top line of the field is parallel to the baseline, and the line between the 1 acre and 3 acre fields is perpendicular to this baseline.
What is the depth of the field?
I have figured this out for the simple case where the angles are both 90 degrees, but even that involved solving a quadratic equation. I didn't expect it to be this hard!
Looks like you need the depth of the field, and where the perpendicular intersects the base. Let's set $L = 764.81'$, $t=8'$, $a = 89.8808^{\circ}$, and $b = 80.6297^{\circ}$. Call your depth (as labeled) $d$, and the base of the right area $x.$
The area of the right section is a trapezoid; we can calculate the area as a rectangle minus a skinny right triangle: $$A_2 = xd - d^2/(2 \tan b).$$
The area of the left section we get the same way, but with another triangle taken out: $$A_1 = (L-x)(d+t) - (d+t)^2/(2 \tan a) - t^2/2.$$
Solve for $x$ in terms of the knowns and $d$ in the first equation, and then solve for $d$ by substituting the expression for $x$ into the second equation. Now that you have a number for $d$, go back to the first equation and get a number for $x$. (Or you can solve the other one first for fun if you like.)
When I ran the cubic equation you get for $d$ in Excel, I got that $d\doteq 226.2019$ feet. Solving for $x$ then gives $x \doteq 578.445$ feet. Hopefully that's close enough (and correct).