Rom

An idea for a biology project might be to study the effect of increasing salinity on the development of freshwater organisms.

I was just reading that this is becoming a problem for North Carolina.

how would i be able to display my results since it will be in an enclosed room.

I don't know what you mean by this.  Do you mean how do you take weight measurements w/o disturbing the system?

You don't.  You just have to go in and take the measurement carefully and accept your interference as a source of uncertainty.

$50 won't get you robots to sit inside the room and do it. :D 

I work with horses.  Our corrals get flooded during rain and at some point evaporation becomes the

main way they dry.  One of the things I have to do is spread the puddles from the low areas out so they'll dry faster.

The experiment should show that evaporation from the 100% sand is fastest, and evaporation from the 100% water is very slow.

This is why we use sand in corrals.

Drainage and evaporation also comes up in all construction, though it's more drainage there.  At least if done correctly.

I suppose there are more exciting projects.

When I judged these things I looked more for good experiment design and data analysis than for bells and whistles.

We'd get projects where the parents clearly funded and helped that were real fancy but the write up showed the kid

had no clue.  I didn't rate those well.  Whereas we'd also get shoestring projects with a clear hypothesis and a

well designed experiment and good write up that I would rate very well.

You have any areas that you'd like to pull your project from?  Any real world problems concern you?

One thing you might try is to compare evaporation rates for various water scenarios.

For example pour a gallon of water into buckets with

a) nothing

b) mostly gravel

c) mostly sand

d) a 50/50 mix

e) a bunch of ordinary sponges

weigh the buckets after you've done this.

Put these buckets into the same area and record the weight of the buckets vs. time

Since the gravel/sand/sponges don't change weight you are essentially measuring the evaporation rate.

Do this all carefully including measurement error.  Grind your data through standard analysis routines to 

come up with the rate +/- your final measurement error.

Present it all cleanly and clearly.

Cheap and easy experiment.  Gives you a chance to do some real data analysis.  This is actually a fairly important real life problem.

Supplementary angles add to 180deg

Interior angles of a triangle add to 180deg

Interior angles of a quadrilateral add to 360deg

what's your budget?

\(f=\dfrac{x-y}{x^4+y^4+6}\)

\(\text{intuition tells me }y=-x\\ \text{plugging this in we get}\\ f = \dfrac{2x}{2x^4+6} = \dfrac{1}{x^3+\frac 3 x}\\ \dfrac{df}{dx} = \dfrac{1}{\left(x^3+\frac 3 x\right)^2}\left(3x^2-\dfrac{3}{x^2}\right)\\ \dfrac{df}{dx}=0 \Rightarrow 3x^2=\dfrac{3}{x^2}\\ x^4=1,~x=\pm 1,~y=-x\\ x=-1,~y=1 \text{ will be a minimum}\\ x=1,~y=-1 \text{ will be a maximum}\)

\(\text{The maximum value is }\dfrac{2}{2+6}=\dfrac 1 4\)

This can all be verified by actually solving \(\nabla f = (0,0)\) for it's real roots.  These turn out to be as mentioned above.

\((AB)^{-1} = B^{-1}A^{-1}\)

\((AB)^{-1}\begin{pmatrix}1\\2\end{pmatrix}=B^{-1}A^{-1}\begin{pmatrix}1\\2\end{pmatrix}\)

\(A\begin{pmatrix}2 \\-1\end{pmatrix}=\begin{pmatrix}1\\2\end{pmatrix} \Rightarrow A^{-1}\begin{pmatrix}1\\2\end{pmatrix} = \begin{pmatrix}2\\-1\end{pmatrix}\)

\((AB)^{-1}\begin{pmatrix}1\\2\end{pmatrix} = B^{-1}\begin{pmatrix}2\\-1\end{pmatrix}=\begin{pmatrix}1\\3\end{pmatrix}\\ \text{(for reasons identical to that of }A)\)

\(\displaystyle \int - \left[\dfrac{1}{\sqrt{1-x^2}}\right]~dx = -\arcsin(x) \text{, not }\arccos(x)\text{ for one}\)

\(\sigma = \text{With sample size of }N \text{ and population standard deviation }\sigma_p\\ \text{the standard error of the mean }= \dfrac{\sigma_p}{\sqrt{N}}\\ \text{A 95% confidence interval for the sample is that interval centered about}\\ \text{the sample mean that contains 95% of the probability mass}\\ \text{This turns out to be the interval }[\mu_s - 1.96 \sigma,~\mu_s + 1.96\sigma]\)

I'll let you plug and chug the numbers but it should be reasonably clear that Bob's sample does look fishy.