Alan

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Anyone who wants to.  There is no guarantee that the answers are correct, so it's important for those asking the questions to check the answers for themselves!

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As Nauseated indicated, the police radar detects the component of the true speed of the car as seen from the direction of the police car. I simply scaled the velocity vectors so the blue one (measured speed, 95 ft/s) occupied the whole distance from car to police car.  I could have drawn both vectors a different length, but they would still have had the same relative length.  The important relationship is: true speed*cos(angle) = measured speed.

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Let f1 be the fraction of isotope 1 and f2 be the fraction of isotope 2.  Then:

f1 + f2 = 1

68.926*f1 + 70.925*f2 = 69.72

Replace f2 in the second equation using the fact that f2 = 1 - f1 from the first equation and solve for f1:

$${\mathtt{68.926}}{\mathtt{\,\times\,}}{\mathtt{f1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{70.925}}{\mathtt{\,\times\,}}\left({\mathtt{1}}{\mathtt{\,-\,}}{\mathtt{f1}}\right) = {\mathtt{69.72}} \Rightarrow {\mathtt{f1}} = {\frac{{\mathtt{1\,205}}}{{\mathtt{1\,999}}}} \Rightarrow {\mathtt{f1}} = {\mathtt{0.602\: \!801\: \!400\: \!700\: \!350\: \!2}}$$

So f1 ≈ 0.603 and f2 ≈ 1 - 0.603 = 0.397

or f1 ≈ 60.3% and f2 ≈ 39.7%

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Here's WolframAlpha's result:

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Melody, this diagram describes the situation as I see it (note: I picture the car driving from left to right rather than bottom to top!):

Note that the red vector is longer than the blue vector.

Imagine that the radar gun was a further 120 ft to the left (i.e. immediately above the initial position of the car), then it would detect zero speed.  If directly in front of the car (i.e. on the road) then it would detect the full speed of the car.  In its specified position it detects a component of the speed.  It can't detect a speed greater than that of the car!

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You can write this as

$$\\ \frac{5\times a}{5}+\frac{2\times a}{5}=-21\\\\\frac{7\times a}{5}=-21\\\\7\times a=-21\times 5\\\\a=\frac{-21\times 5}{7}\\\\a=-3\times 5\\\\a=-15$$

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No, a palindrome reads the same backwards as forwards.  For example, 1991 is a palindrome.

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The measured speed would be greater if the radar were measuring the speed of the car head on, so it should be that v*cos(θ) = 95, where v is the car's speed and θ is the angle between the road and the line from the initial position of the car to the police car.

θ = tan^-1(30/120) ≈ 14°, so car speed v ≈ 95/cos(14°) = 97.9 ft/s

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With this small angle, sines and tans are largely irrelevant!  The diameter is just the angle multiplied by the distance - as long as the angle is expressed in radians.

$${\mathtt{diameter}} = {\frac{{\frac{{\mathtt{34}}}{{\mathtt{60}}}}{\mathtt{\,\times\,}}{\mathtt{\pi}}}{{\mathtt{180}}}}{\mathtt{\,\times\,}}{\mathtt{92\,919\,800}} \Rightarrow {\mathtt{diameter}} = {\mathtt{918\,995.321\: \!833\: \!621\: \!997\: \!390\: \!4}}$$

diameter ≈ 9.19*10⁵ miles

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