Alan

If height, h = 20 - 16t² + 32t, then upward velocity, v = dh/dt = -32t + 32.  At its highest point the speed is zero, so this happens when -32t + 32 = 0 or when t = 1.  At this time the height is hmax = 20 -16*1² +32*1 = 36 feet

Most of them are, but:

16.  Metal feels colder than wood because the rate it draws heat from your hand is greater than that of wood.  Because it's the rate of transfer of energy that is important here, it is the specific heat capacity (larger for wood than metal, so rate of change of temperature is smaller) that matters most.

25. Red light is refracted least.

30.  Atoms radiate photons when they undergo a transition from a high energy state to a low one.

I can't read question 7 in the last image!

Take logs of both sides:  ln(44118) = ln(3^b)

From a property of logs we have ln(3^b) = b*ln(3) so ln(44118) = b*ln(3); hence b = ln(44118)/ln(3)

$${\mathtt{b}} = {\frac{{ln}{\left({\mathtt{44\,118}}\right)}}{{ln}{\left({\mathtt{3}}\right)}}} \Rightarrow {\mathtt{b}} = {\mathtt{9.734\: \!665\: \!497\: \!315\: \!512\: \!4}}$$

Because you have arithmetic mod 65537, you could find other answers by adding multiples of 65537 on to 44118 first.  For example

$${\mathtt{b2}} = {\frac{{ln}{\left({\mathtt{44\,118}}{\mathtt{\,\small\textbf+\,}}{\mathtt{65\,537}}\right)}}{{ln}{\left({\mathtt{3}}\right)}}} \Rightarrow {\mathtt{b2}} = {\mathtt{10.563\: \!412\: \!108\: \!297\: \!925\: \!2}}$$

$$\left({{\mathtt{3}}}^{\left({\frac{{ln}{\left({\mathtt{44\,118}}{\mathtt{\,\small\textbf+\,}}{\mathtt{65\,537}}\right)}}{{ln}{\left({\mathtt{3}}\right)}}}\right)}\right) {mod} \left({\mathtt{65\,537}}\right) = {\mathtt{44\,117.999\: \!999\: \!999\: \!96}}$$

(A small numerical error creeping in there!)

$$This can be written as $$2^2\times2^{n-1}=(2^4)^n$$\\\\
or $$2^{n+1}=2^{4n}$$\\\\
In other words $n+1=4n$ or $n=\frac{1}{3}$$$

if you mean log to the base 10, then 0.123 = -logT means that T = 10^-0.123 or 

$${\mathtt{T}} = {{\mathtt{10}}}^{-{\mathtt{0.123}}} \Rightarrow {\mathtt{T}} = {\mathtt{0.753\: \!355\: \!563\: \!733\: \!717\: \!4}}$$

I think the MIN function would return the smallest number in the list (which would be 1 from 5 3 3 1 1), but the questioner is asking for the number that occurs least frequently in the list(which would be 5 from 5 3 3 1 1).

I'm not aware of a simple function in Excel that will do this (though that doesn't necessarily mean there isn't one!).

For your expression to be zero you must have the term in brackets in the numerator to be zero; so:

pi²*r⁶ - 64800 = 0 or r = 64800^1/6/pi^1/3

$${\mathtt{r}} = {\frac{{{\mathtt{64\,800}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{6}}}}\right)}}{{{\mathtt{\pi}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}}} \Rightarrow {\mathtt{r}} = {\mathtt{4.327\: \!255\: \!548\: \!039\: \!541\: \!4}}$$

Here it is:

No, it isn't a power of 3:   3^4 = 81 and 3^5 = 243

But it is a multiple of 3:  3*75 = 225

I can't see the images either.