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# What Should My Sister Put on Her Wedding Registry?

We have all the standard stuff that you need for a house too. I've already owned 3 houses and have moved into a bigger house every time I buy a new one so of course, I have enough stuff. She had her own apartment for the last 3 years. However, she decided she wanted all new stuff so that's what we asked for on the registries. Towels, sheets, comforter, mattress cover, dishes, cutlery, appliances, etc. We had a few nontraditional items on the registry too. Dog bed for our dog (purchased), a new gas grill (yes, somebody bought it), a motorized cooler on wheels (got that too!), a microwave (got it) and stuff like that. Those are just some ideas you can registered for. Go shopping in these stores that have registries and just pick out stuff you want. That's what we did. 1. maintenance schedule tracking software

You mention Strava: they do that for you. If you register components and parts of your bike like a chain it will show the mileage from the registering date. It might a bit problematic if you have two sets of wheels (like one for rain and another for sunny days)that you use alternatively

2. How many wheels are on a boat?

Depends on your definition of a wheel and what kind of boat.By the most common definition, there are no wheels on my sailboat. But you could call a "block" with a rotating pulley in it a wheel. This would bring the sailboat up to 12 wheels.If you mean "steering wheels" or the wheel for the helm, again my sailboat has none. It uses a tiller. Some sailboats have just one wheel for the helm. Some of the large sailing vessels had many, because the power needed to hold the helm was far more than one person could exert and power steering did not exist. 3. Road Shape for Square Wheels

Let Choose a coordinate system so that \$Q\$ is constrained to move on the line \$y=0\$. This means \$y_Q = 0\$ identically.Parameterize \$P\$ by arc-length \$s\$. We will assume the wheel is rolling from left to right without sliding. Furthermore, at \$s = 0\$, \$Q\$ is located at origin and lies directly above \$P\$. More precisely,\$\$(x_Q(0), y_Q(0)) = 0,quadtext and quad begincases ( x_P(0), y_P(0) ) = (0,-b) ( x'_P(0),y'_P(0)) = (1,0) endcases\$\$After we roll the wheel for a distance \$s\$, \$P\$ moved to \$(x_P(s), y_P(s))\$. The tangent vector and upward normal vector of the bump at that point equals to \$t = (x'_P(s),y'_P(s))\$ and \$n = (-y'_P(s), x'_P(s))\$. Since the wheel is rolling without sliding with respect to the bump. One can reach \$Q\$ from \$P\$ by a move along direction \$n\$ for a distance \$b\$ followed by a move along direction \$-t\$ for distance \$s\$. This leads to following ODE\$\$0 = y_Q(s) = y_P(s) - s y'_P(s) b x'_P(s)\$\$Together with the constraint \$x'_P(s)^2 y_P'(s)^2 = 1\$ and given initial conditions, one find:\$\$ begincases x_P(s) = bsinh^-1fracsb y_P(s) = -sqrts^2b^2 endcases quad iff quad begincases x_P(s) = bt y_P(s) = -bcosh t endcases quadtext where quad t = sinh^-1fracsb \$\$ Let \$t_0 = sinh^-1(1) = log(1sqrt2)\$. The two endpoints of the bump corresponds to \$s = pm b iff t = pm t_0\$. At those points, \$y_P(s)\$ reaches its lowest value \$-sqrt2b\$. The area under the bump (but above this \$y\$) is given by the formula:\$\$beginalignint_-b^b (y_P(s)sqrt2b)x'_P(s) ds &= b^2 int_-t_0^t_0 (sqrt2 - cosh t) dt = 2b^2 left(sqrt2t_0 - sinh t_0

ight) &= fraca^22 left(sqrt2log(1sqrt2) - 1

ight) endalign \$\$ UpdateAbout how to solve the ODE, differentiate both equations by \$s\$, one get \$\$ begincases y - s y' b x' &= 0 y'^2 x'^2 &= 1 endcases quadstackrelfracddsLongrightarrowquad begincases -sy'' bx'' &= 0 y'y'' x'x'' &= 0 endcases \$\$ Comparing coefficients for \$x''\$ and \$y''\$, we find there is a function \$lambda(s)\$ such that \$\$ (x',y') = ( blambda, -slambda ) \$\$Since we want \$x' > 0\$, the normalization condition \$x'^2 y'^2 = 1\$ fixes \$lambda\$ to \$frac1sqrts^2b^2\$. Integrate above equation by \$s\$ will give us the desired solution.

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