Question 1019740
the area of a triangle is equal to 1/2 * base * height.


let a = area and b = base and h = height and the formula becomes a = 1/2 * b * h


you are given that the height is equal to 4 inches greater than twice the base.


since h = height and b = base, the formula for that would be h = 2b + 4


in the formula of a = 1/2 * b * h, replace h with 2b + 4 to get:


a = 1/2 * b * (2b + 4)


simplify to get a = 1/2 * 2b^2 + 4b


simplify further to get a = b^2 + 2b


since you want the area to be no more than 168 square inches, then you get a <= 168.


since a = b^2 + 2b, then you get b^2 + 2b <= 168 after you replaced a with it's equivalent value of b^2 + 2b.


so the formula you want is b^2 + 2b <= 168.


if you subtract 168 from both sides of that equation, you will get b^2 + 2b - 168 <= 0


to find the zero crossing points, set b^2 + 2b - 168 = 0.


you will find that the roots of that equation are b = -14 or b = 12.


you have 3 intervals on your graph you need to examine.


they are:


b < -14
b >= -14 and b <= 12
b > 12


you will find that the graph is less than or equal to 0 in the interval where b >= -14 and b <= 12


the graph looks like this:


on the graph, the variable x is used in place of b.


<img src = "http://theo.x10hosting.com/2016/021502.jpg" alt="$$$" </>


since the length of the base can't be less than or equal to 0, than you have that 0 < b <= 12.


when b = 12, h = 24+4 = 28, and a = 1/2 * b * h = 1/2 * 12 * 28 = 168.


when b = 11, h = 22+4 = 26, and a = 1/2 * b * h = 1/2 * 11 * 26 = 143.


when b = 0, the triangle collapses into a straight line.
it's debatable you can still call it a triangle, but if you think about sin(90) = 1, that only occurs when the triangle has collapsed, so you might think yes, but i've seen other sources that say no.  


i'm assuming no in this presentation which is why i say the base can't be 0.