Alan

√3 is irrational.  This means it cannot be expressed exactly by the ratio of two whole numbers.  That is, it cannot be expressed exactly as a fraction.

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In general log_na - log_nb = log_n(a/b) so log₄(128) - log₄(8) = log₄(128/8) = log₄(16)

Also, when n^a = b then a = log_nb so log₄(16) = 2 (because 4² = 16)

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cosine is positive in the first and fourth quadrants, so the angle whose cosine is 1/√5 is not just 1.107, but also -1.107.  However, this angle can also be expressed as 2pi - 1.107.  Using 2pi - 1.107 should give you the other solution in the range 0 to pi.  (Most calculators/software will just give one value when asked for cos^-1 .)

See image:

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Electrostatic force is inversely proportional to the square of the distance between the two balloons, so if this distance is tripled the force will be 9 times smaller, namely:   0.32/9 N

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I get a slightly different answer for the second part:

prob of matching 3 white b***s  = 3/20 * 2/19 * 1/18 = 6/(20*19*18)

prob of matching 1st and 2nd but not 3rd = 3/20 * 2/19 * 17/18 = 6*17/(20*19*18)

prob of matching 1st and 3rd but not 2nd = 3/20 * 17/19 * 2/18  = 6*17/(20*19*18)

prob of matching 2nd and 3rd but not 1st = 17/20 * 3/19 * 2/18  = 6*17/(20*19*18)

prob of matching blue ball = 1/10

Overall probability = 6*(1+3*17)/(20*19*18) +1/10 = 83/570 ≈ 0.146

I think Chris's probability (23/228) is that of getting all three white b***s or the blue ball, rather than at least two white b***s or the blue ball.

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An infinite number of √2's added together is infinite, so the square root of that is also infinite.

Is that what you really meant?

Form the standard error: s = 73.12/√82

Assuming a 2-sided confidence interval then the upper limit is at a cumulative probability value of 96.92+(100 - 96.92)/2 = 98.46%

Using the t distribution (because we only know the sample standard deviation, not the true population value) this cumulative probability occurs (with 81 degrees of freedom) when t = 2.198 standard errors.

Hence upper limit = 49.75+2.198*73.12/√82

$${\mathtt{ul}} = {\mathtt{49.75}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{2.198}}{\mathtt{\,\times\,}}{\mathtt{73.12}}}{{\sqrt{{\mathtt{82}}}}}} \Rightarrow {\mathtt{ul}} = {\mathtt{67.498\: \!307\: \!504\: \!130\: \!928\: \!9}}$$

or at approximately 67.5

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tan(angle of depression) = height/horizontal distance

so horizontal distance = height/tan(angle of depression)

$${\frac{{\mathtt{1\,700}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{11}}^\circ\right)}}} = {\mathtt{8\,745.741\: \!827\: \!136\: \!861\: \!007\: \!1}}$$

horizontal distance ≈ 8746 ft 

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The intersection with the x-axis occurs when f(x) = 0, so

(3/2)x - 3 = 0

(3/2)x = 3

(1/2)x = 1

x = 2

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