Forming Functions from Verbal Descriptions

[soulofeternity]

I really have no idea where to start this problem at all and I don't have a clue as to how they are getting the answer.

Question-Express the area A of a 30-60-90 triangle as a function of the length h of the hypotenuse.

Answer (I can't type how it looks in the book)- (h squared times the square root of three) divided by 8.

 

[skeeter]

you should know the relationships between the three sides of a 30-60-90 triangle ...

hypotenuse is twice the length of the short leg

long leg is \(\displaystyle \sqrt{3}\) times the short leg.

so ... if the hypotenuse is h, then the short leg is h/2, and the long leg is (h/2)\(\displaystyle \sqrt{3}\).

area = (1/2)(length of the short leg)(length of the long leg)

substitute and simplify.

 

[soulofeternity]

I'm just terrible with stuff like this. . . Thank you for explaining it to me! ^.^

 

[soroban]

Hello, soulofeternity!

Let me try it with a diagram . . .

Express the area \(\displaystyle A\) of a 30-60-90 triangle as a function of the length \(\displaystyle h\) of the hypotenuse.

Answer: \(\displaystyle \,\frac{h^2\sqrt{3}}{8}\)

You're expected to know that a 30-60-90 triangle has sides in the ratio: \(\displaystyle \,1\,:\,\sqrt{3}\,:\,2\)

                      *
            2k    *   |
              *       | k
          * 30°       |
      * - - - - - - - *
            k√3

Since \(\displaystyle 2k\,=\,h\), then: \(\displaystyle \,k\,=\,\frac{h}{2}\,\)and \(\displaystyle \,k\sqrt{3}\,=\,\frac{h\sqrt{3}}{2}\)

We have a right triangle with: \(\displaystyle \tex{base}\,=\,\frac{h\sqrt{3}}{2}\,\) and \(\displaystyle \,\text{height}\,=\,\frac{h}{2}\)


Therefore: \(\displaystyle \,\text{Area} \:=\:\frac{1}{2}\text{(base)(height)} \:=\:\frac{1}{2}\left(\frac{h\sqrt{3}}{2}\right)\left(\frac{h}{2}\right)\:=\:\L\frac{h^2\sqrt{3}}{2}\)