a.) A = P+P*r*t for r
Does anyone else get r = (A/P-1)/t ?
b.) d = (sqrt(4*R^2-C^2))/2 for C
Does anyone else get C = sqrt((-2d)^2)/4)+R^2
c.) 1/f = 1/q+1/p for q
Does anyone else just get f^-1-p^-1=q^-1?
a) Your answer is correct, but I would have done it this way. Less complex:
$\displaystyle A=P+Prt$
$\displaystyle A-P=Prt$
$\displaystyle r=\frac{A-P}{Pt}$
b) You lost me on this one.
$\displaystyle d=\frac{\sqrt{4R^2-C^2}}{2}$
$\displaystyle d^2=\frac{4R^2-C^2}{4}$
$\displaystyle 4d^2=4R^2-C^2$
$\displaystyle 4d^2-4R^2=-C^2$
$\displaystyle 4R^2-4d^2=C^2$
$\displaystyle C^2=4R^2-4d^2$
$\displaystyle C=\sqrt{4(R^2-d^2)}$
$\displaystyle C=2\sqrt{R^2-d^2}$
c) I don't see where you solved for q in your attempt.
$\displaystyle \frac{1}{f}=\frac{1}{q}+\frac{1}{p}$
Multiply through by the LCD of fqp
$\displaystyle qp=fp+fq$
$\displaystyle qp-fq=fp$
$\displaystyle q(p-f)=fp$
$\displaystyle q=\frac{fp}{p-f}$