The initial value is 4 and the rate of change is -3/4
Answer:
you can type it into a calculator you know. it gives the right answer
5000 young adults to visit the mall because the author of young adult books want to know what his/her readers prefer. So targeting young adults in a survey seems the best option as that is the type of audience her books are mainly for.
An equilateral triangle isn't a right triangle, so the Pythagorean Theorem doesn't apply directly.
But if we divide an equilateral triangle with side <em>s</em> in half along an altitude, we get two congruent right triangles each with hypotenuse <em>s </em>and one leg <em>s/2.</em>
The other leg is the altitude <em>h</em> we divided along. By the Pythagorean Theorem
![( \frac s 2)^2 + h^2 = s^2](https://tex.z-dn.net/?f=%28%20%5Cfrac%20s%202%29%5E2%20%2B%20h%5E2%20%3D%20s%5E2)
![\frac{s^2}{4} + h^2 = s^2](https://tex.z-dn.net/?f=%5Cfrac%7Bs%5E2%7D%7B4%7D%20%2B%20h%5E2%20%3D%20s%5E2)
![h^2 = \frac{3}{4} s^2](https://tex.z-dn.net/?f=h%5E2%20%3D%20%5Cfrac%7B3%7D%7B4%7D%20s%5E2)
![h = \dfrac{\sqrt{3}}{2} s](https://tex.z-dn.net/?f=h%20%3D%20%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20s)
The area <em>A</em> of our original equilateral triangle is now
![A= \frac 1 2 sh = \frac 1 2 s \left( \dfrac{ \sqrt{3}}{2} s \right) = \dfrac{\sqrt{3}}{4} s^2](https://tex.z-dn.net/?f=A%3D%20%5Cfrac%201%202%20sh%20%3D%20%5Cfrac%201%202%20s%20%5Cleft%28%20%5Cdfrac%7B%20%5Csqrt%7B3%7D%7D%7B2%7D%20s%20%5Cright%29%20%3D%20%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B4%7D%20s%5E2)
So the area of our equilateral triangle of side 5 is
![A = \dfrac{\sqrt{3}}{4} (5)^2 = \dfrac{25}{4} \sqrt{3} \approx 10.8](https://tex.z-dn.net/?f=A%20%3D%20%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B4%7D%20%285%29%5E2%20%3D%20%5Cdfrac%7B25%7D%7B4%7D%20%5Csqrt%7B3%7D%20%5Capprox%2010.8)
Answer: 10.8
I don't like ruining a nice exact answer with an approximation, but the question asks for one.
We could have done the above steps with 5 in particular instead of <em>s </em>but I prefer to do the general solution and substitute at the end.
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Since you read this far, I will give you the secret formula for the area S of a triangle given <em>squared</em> sides A, B, C. Your teachers won't even know this one.
![16S^2 = 4AB-(A+B-C)^2](https://tex.z-dn.net/?f=16S%5E2%20%3D%204AB-%28A%2BB-C%29%5E2)
Let's try it out for
an equilateral triangle:
![16S^2 = 4A^2 -A^2 = 3A^2](https://tex.z-dn.net/?f=16S%5E2%20%3D%204A%5E2%20-A%5E2%20%3D%203A%5E2)
![S = \sqrt{\dfrac{3}{16}} A = \dfrac{\sqrt{3}}{4} s^2 \qquad\checkmark](https://tex.z-dn.net/?f=S%20%3D%20%5Csqrt%7B%5Cdfrac%7B3%7D%7B16%7D%7D%20A%20%3D%20%5Cdfrac%7B%5Csqrt%7B3%7D%7D%7B4%7D%20s%5E2%20%5Cqquad%5Ccheckmark)