Alan

Here's a similar derivation, but makes explicit use of partial fractions:

.

Hi Melody!

Perhaps I should have included the following extra step:

 \(\frac{10}{3\sqrt5}\rightarrow\frac{10\sqrt5}{3\times\sqrt5\times\sqrt5}\rightarrow\frac{10\sqrt5}{3\times5}etc\)

i.e. you multiply the expression by 1 in the form \(\frac{\sqrt5}{\sqrt5}\)  Then you multiply the two \(\sqrt5\)s on the bottom together.

Hope this explains it adequately for you.

.

If the equation is x = (y - 1)/y then Guest #1 is correct.  

If it is  x = y - 1/y  then it is quadratic in y and we have:   \(y = \frac{x\pm \sqrt{x^2+4}}{2}\)

.

Like so:  \(\frac{10}{3\sqrt5}\rightarrow\frac{10\sqrt5}{3\times5}\rightarrow\frac{2\sqrt5}{3}\)

.

Part (a)

f(x) = (ax + b)/(x + c)

f(infinity) = - 4  so  - 4 = a       (limit as x goes to infinity of (ax+b)/(x+c) is a)   so a = -4

f(3) = infinite so  3 + c = 0     so c = -3

f(1) = 0        so  (a + b)/(1 + c) = 0   

                          (-4 + b)/(1 - 3) = 0      so b = 4

hence f(x) = (-4x + 4)/(x - 3)      or    f(x) = 4(1 - x)/(x - 3)

I'll leave you to do part (b).

A regular hexagon is comprised of 6 equilateral triangles. Each side is s = p/6

Area of 1 triangle:  A1 = (1/2)*s*sqrt(s^2 - (s/2)^2) → (1/2)*s^2*sqrt(3/4) → (1/72)*p^2*sqrt(3/4) 

Area of hexagon;  A = 6A1 → (6/72)*p^2*sqrt(3/4) → (1/12)*p^2*sqrt(3/4) → (1/24)*p^2*sqrt(3)

Hence p^2/A = 24/sqrt(3) → 8sqrt(3)

.

Enter sin(8+25/60+30/3600)  

(make sure the calculator is in degree mode).

.

Volume of a cylinder = pi*(diameter/2)^2*height

Volume = pi*(225*2214/2)^2*2214  ft^3

I'll leave you to crunch the numbers!

.