There are two approaches to this problem.  We are able to calculate the angle by using the Sine Law.  The Sine Law states:

So we can set  and solve accordingly for angle .

Our other option is to use the area formula we have been but altering it to correspond to angle .  We would draw our vertical line down from the vertex as shown below and our formula would be in the form .

Even though the formula  is using sides and angle , this is a general formula and can be used with any angle in the triangle.  Since we are now working with an obtuse angle rather than an acute angle, we need to do some more work to get the logic right.

Using the figure above, to be able to label the sides we are using correctly, we extend the original triangle horizontally past the obtuse angle and draw a vertical line down from the top vertex to form a right angle.  This vertical line is .  Angle  for the supplementary (orange) triangle is .  Using the fact that , we can set up our formula to be the following:

 (either angle can be used and I will show this to be true)

  (Using from the original triangle)

This shows that when using this formula for an obtuse angle, you can use either the supplementary angle you made, or the original.  It is always helpful to draw this supplementary triangle in order to be able to visualize and understand logically how the formula is working for obtuse angles as well.

In order to use this formula you need to either the height (which can be used to find the angle) or the angle (which can be used to find the height).

The formula for the area of a triangle is (base)(height).  Since this is an obtuse triangle we need to break it into two right triangles by drawing the line down from the vertex perpendicular to the opposite side.

Now that we have two right triangles we can solve for the area of this triangle.  Notice our base is and our height is .  Plugging into the formula for the area of a triangle gives us .  Most of the time, we will not have an exact length for , but we may have, or will be able to solve for, the lengths of  and the angles .  Using our  relationship for right triangles, we know .  In this case we will use  as our angle.  So

We can plug  in for in our formula for area and we are left with .

We must begin by finding the area first.  To find the area we begin by using our equation and recall that .  We will let:

So our area is 

How to find : , , 

How to find : , , 

To find , we know the angles of a triangle should add up to 180 so:

To solve for , we first need to find the area using our formula , considering the triangle .  We will begin by labeling our variables:

To find the area, we plug the values for the variables into the formula.

Knowing that our area is 24, we can now solve for .  Recall that:

It follows that:

   (because )

  (plugging in our values for area and )

Recall that  for right triangles.  When we draw a line down from the vertex to the opposite side this creates two right triangles within the original obtuse triangle.  If we are solving for , then  becomes our hypotenuse and  becomes the opposite side we are working with.  Therefore  .

Using the formula , we will let  and 

Following from the figure above, we draw an auxiliary line from the vertex perpendicular to the opposite side.  We will call this line  for height and will label the new vertex .  The formula we will use is .  First we begin by labeling our variables.  We will let  be the base of our triangle and  be the hypotenuse of the triangle formed by .  The values for the variables are as follows:

Plugging these into our formula, it follows that:

Using the formula , we will let  and .

 (because )

Then we must account for each square foot costing $3.

The answer is $162.