(1) Two circles of radius 1 and 4 units are tangent
to each other externally. the length of the common
tangent line between the points of tangency is
closest to which of the following?
(a) 4.8 (b) 4.5 (c) 3.9 (d) 3.6
Draw Circle A with radius 1 externally tangent to
Circle B with radius 4.
Let CD be their common tangent, where CD is tangent
to circle A at C and tangent to circle B at D.
Connect the two centers with segment AB
Draw radii AC and BD.
AC and BD are parallel because they are both
perpendicular to CD.
Draw AE parallel and equal to to CD where E is on BD.
Here is the figure, I can't put in the letters here but
you can on your paper:
ABE is a right triangle.
AB is its hypotenuse and equals 1+4 or 5
AC=DE=1, and BD = 4, so BE = 3
By the Pythagorean theorem
BE² + AE² = AB²
3² + AE² = 5²
9 + AE² = 25
AE² = 16
AE = 4
and CD = 4 since CD=AE and AE=4
Therefore the common tangent line is 4.
So the answer is (c) because 3.9 is
nearest of the choices to 4.
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(2)If the diameter of a cylindrical jar is
increased by 25% without altering the volume
then its height must decrease by:
(a) 10% (b)25% (c)36% (d)50%
Let the answer be p%
Volume of a cylinder = p(radius)²(height)
Radius = Diameter/2
Volume of cylinder = p(diameter/2)²(height) = p(diameter)²height/4
Diameter before = d
Diameter after = d + .25d = 1.25d
Height before = h
height after = h - .01ph = h(1-.01p)
Volume before = p(d)²h/4
Volume after = p(1.25d)²[h(1-.01p)]/4
Volume after = Volume before
p(1.25d)²[h(1-.01p)]/4 = p(d)²h/4
p(1.25)²d²h(1-.01p)/4 = pd²h/4
p(1.5625)d²h(1-.01p)/4 = pd²h/4
Multiply both sides by 4
p(1.5625)d²h(1-.01p) = pd²h
Divide both sides by pd²h
1.5625(1-.01p) = 1
1.5625-.015625p = 1
-.015625p = -.5625
p = (-.5625)/(-.015625)
p = 36
So the height decreased by 36%, choice (c)
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Interesting note: for some reason, statistics have shown that
multiple-choice test-makers make the correct choice
(c) more often than (a), (b), or (d). So if a student does
not know an answer and cannot eliminate (c), and must guess,
they should always guess (c), because their chances of guessing
correctly is greater. You will notice that both correct choices
here were indeed (c), bearing out the theory. :-)
Edwin
