A stormwater drain has the shape illustrated. Determine the angle β where the left hand side meets with the bottom of the drain. See attached image for shape and dimensions

Steps to Solve

1. Analyze the Shape and Identify Relevant Dimensions

We have a stormwater drain with given dimensions: top width 5m, bottom width 2m, height 3m, and one angle of 100 degrees. We need to find the angle $\beta$.

2. Calculate the Horizontal Distance for the 100-degree Side

Let's find the horizontal distance corresponding to the side with the 100-degree angle. We can use the tangent function, but first, we need the angle inside the right triangle formed by the height and this horizontal distance. Since the angle at the bottom is 100 degrees, the angle inside the triangle next to the 3m height is $180 - 100 = 80$ degrees. This angle is complementary to the one in the triangle, thus we have $90 - 80 = 10$ degrees, therefore the angle within the right triangle is $100 - 90= 10$ degrees. Let $x$ be the horizontal length of the right-hand side.

$$ \tan(10^\circ) = \frac{3}{x} $$

$$ x = \frac{3}{\tan(10^\circ)} \approx \frac{3}{0.1763} \approx 17.016 $$

Note: the horizontal distance calculation can be done using $80$ degrees with the following: $$ \tan(80^\circ) = x/3 $$

$$ x = 3*\tan(80^\circ) \approx 3*5.671 \approx 17.013 $$

3. Calculate the Horizontal Distance for the Angle Beta Side

Let call the horizontal distance of the beta side $y$. The total top width is 5m and the bottom width is 2m. Therefore, the difference between the top and bottom widths gives the combined length of $x + y$. So: $5 - 2 = x + y = 3$ And we found that $x \sim 17.0$, which clearly does not make sense Therefore, there is likely a typo in the diagram. I will assume that the angle $\beta$ and the $100^\circ$ angle occur at the top of the figure, meaning $\beta$ is on the top left. If that is the case, the problem drastically changes.

Update There must be a horizontal component on BOTH sides. If there isn't, then the beta angle is oriented incorrectly. I will rotate Beta so one is the top and the other is the bottom of the trapezoid.

If we let $y$ be the length of the left side. $5 = 2 + x + y$ where $x$ and $y$ were previously defined. $x + y = 3$

Since we know the height of the side is $3$, we can use the tangent function to express $y$: $\tan(\theta) = 3/y$

The angle that is supplementary to $\beta$ is $180 - \beta$. Therefore $\theta = 180 - \beta - 90 = 90 - \beta$ There is no other information, therefore we cannot continue to solve the issue.