Rom

without units the question makes no sense.

6 what?  cubic centimeters or cubic miles?

54 grams? or metric tons?

\(\mbox{If }y \mbox{ varies directly with }x \mbox{ then }y = c x \\ \mbox{If }y \mbox{ varies indirectly with }x \mbox{ then }y = \dfrac c x \\ \mbox{For some number }c\)

\(mass = volume \times density \\ mass = 2 ml \times 8.5 g/ml = 2\times 8.5 g = 17g\)

The object will displace it's own volume in the water.  So the difference between the initial and final levels is the volume of the object.

V = 26.5 - 15 = 11.5ml

oops, I didn't answer the question on this one.  Not sure why I thought it wanted total length when it clearly specifies the angle.

Interesting problem.

We treat the staircase as a space curve.

\(r(t)=\{7 \cos(2 \pi t), 7 \sin(2 \pi t), 37 t\},~~~~0\leq t \leq 1\)

To find the length of the curve we integrate the norm of the time derivative of the curve.

\(L = \displaystyle{\int_0^1}\|\dot{r}\|~dt\)

\(\dot{r}=\{-14\pi \sin(2 \pi t), 14\pi \cos(2\pi t), 37\}\\ \|\dot{r}\| = \sqrt{(14\pi)^2+37^2} \\ L=\displaystyle{\int_0^1}\sqrt{(14\pi)^2+37^2}~dt = \sqrt{(14\pi)^2+37^2}\approx 57.48 ft\)

\(\mbox{If you have a quadratic polynomial }a x^2 + b x + c \mbox{ the roots of this are given by}\\ r_{1,2}=\dfrac{-b\pm\sqrt{b^2-4 a c}}{2a} \\ \mbox{The argument of the square root in this expression }(b^2 -4 a c) \mbox{ is called the discriminant.}\)

\(\mbox{It's called this because it's value discriminates between real and complex roots.}\\ \mbox{if it is non-negative then the roots of the quadratic will be real. If it is negative} \\ \mbox{they will be a complex conjugate pair.}\)

\((-5)\times \left(-\dfrac 2 3\right) = \dfrac{(-5)(-2)}{3}=\dfrac{10}{3} = 3\dfrac 1 3\)

\(7\% = \dfrac {7}{100} \mbox{ so} \\ 7\% \mbox{ of }40 = \dfrac{7}{100}\times 40 = 2.8\)

\(2x-1.5 = 31.83 \\ 2x = 33.33 \\ x = 16.665\)