As soon as I read this, the words "law of cosines" popped
into my head. I don't have a good intuitive feeling for the
law of cosines, but I went and looked it up (you probably
could have done that), and I found that it's exactly what
you need for this problem.
The "law of cosines" relates the lengths of the sides of any
triangle to the cosine of one of its angles ... just what we need,
since we know all the sides, and we want to find one of the angles.
To find angle-B, the law of cosines says
b² = a² + c² - 2 a c cosine(B)
B = angle-B
b = the side opposite angle-B = 1.4
a, c = the other 2 sides = 1 and 1.9
(1.4)² = (1)² + (1.9)² - (2 x 1 x 1.9) cos(B)
1.96 = (1) + (3.61) - (3.8) cos(B)
Add 3.8 cos(B) from each side:
1.96 + 3.8 cos(B) = 4.61
Subtract 1.96 from each side:
3.8 cos(B) = 2.65
Divide each side by 3.8 :
cos(B) = 0.69737 (rounded)
Whipping out the
trusty calculator:
B = the angle whose cosine is 0.69737
= 45.784° .
Now, for the first time, I'll take a deep breath, then hold it
while I look back at the question and see whether this is
anywhere near one of the choices ...
By gosh ! Choice 'B' is 45.8° ! yay !
I'll bet that's it !
(126) times (1/2 to the x/11 power) = 15% times 126
(1/2) to the x/11 power = 0.15
Take the log of both sides :
(x/11) times log(1/2) = log(0.15)
Multiply both sides by 11 :
'x' times log(1/2) = 11 x log(0.15)
Divide both sides by " log(1/2) " :
x = 11 x log(0.15)/log(1/2) = <em><u>30.107 years</u></em> (rounded)
That's the time it takes for <em><u>any-size</u></em> sample of this substance
to decay to 15% of its original size.
Dilation is scale factor, like a dilation scale factor of 2
