Alan

I suspect this is meant to be \(\frac{2.72\times10^{22}}{6.02\times10^{23}}\rightarrow \frac{2.72\times10^{-1}}{6.02}\rightarrow0.4518...\times10^{-1}\rightarrow 0.04518...\)

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sqrt(6^2+3^2) → sqrt(45) → sqrt(5*9) → sqrt(5*3^2) → 3sqrt(5)

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The probability of a specific 5 bills containing errors is (0.6^5)*(0.4^5)

There are ncr(10,5) ways of choosing those 5 bills  ( ncr(10,5) = 10!/(5!*5!) )

so the overall probability is ncr(10,5)*(0.6^5)*(0.4^5)

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You should be able to find the item in no more than 6 tries.

First ask if the item is above half-way.  Whatever the answer you have reduced the search space so that you now only have 16 items to search.  Whichever 16 it is, choose the middle of them and ask if the item is above or below this position.  Now the answer will reduce the search space so you only have 8 items to search.  Continue in this way until there is only one possibility left.

No.  1.33333..... etc is one and a third.   0.33333... is 1/3

To see this let x = 0.3333........

Then 10x = 3.333333

so 10x - x = 3

       9x = 3

         x = 3/9

         x = 1/3

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Vertical motion.  Distance given by:  s_vert = 11.5*sin(35°)*t - (1/2)*9.81*t²

 This is zero at times t = 0 and t = 2*11.5*sin(35°)/9.81 seconds.  The latter time is the time at which the rock comes back to the ground.

Horizontal motion.   Distance given by:  s_horiz = 11.5*cos(35°)*t

So, total horizontal distance travelled = 11.5*cos(35°)*2*11.5*sin(35°)/9.81 metres

I'll leave you to crunch the numbers.

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p(0g) = 1/2.               (Probability of no girls)

p(1g) = 1/2^2            (Probability of exactly 1 girl)

p(2g) = 1/2^3.           (Probability of exactly 2 girls)

p(3g) = 1/2^3.           (Probability of exactly 3 girls)

Expected number of girls = 0*1/2 + 1*1/2^2 + 2*1/2^3 + 3*1/2^3 → 7/8

p(0b) = 1/2^3

p(1b)  = 1/2^3 + 1/2^2 + 1/2 → 7/8

Expected number of boys = 0*1/2^3 + 1*7/8 → 7/8

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64ft/s is correct.

Note:  In part 1 my hint contains an error.  \(\frac{du}{dt}=\frac{s}{\sqrt{s^2+1800^2}}\frac{ds}{dt}\) is the correct expression for the rate of change of u.  The 2 shouldn't be there!

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1.  Here's a hint for part 1 Max.

I'm sure you can take it from here.

2.  Here's part 2.

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