The numbers a, b and c satisfy a+b+c = 0 and abc = 78. What isthe value of (a+b)(b+c)(c+a)?
1 answer:
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<h3>Given</h3>
72 blue, 42 red beads
beads are used to make identical necklaces
<h3>Find</h3>(a) the greatest number of necklaces that can be made
(b) the number of each color bead in each necklace
<h3>Solution</h3>You can write and factor the equation
... necklaces = 72 blue + 42 red
... necklaces = 6(12 blue + 7 red)
where 6 is the greatest common factor (GCF) of 72 and 42.
(a) You can make up to 6 identical necklaces. 6 is the largest common factor of 72 and 42. If you were to try to make more, they could not be identical.
(b) Each necklace can consist of 12 blue and 7 red beads. These are the numbers obtained when the total bead count is divided into 6 equal groups.
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There are several ways you can find the GCF of two numbers. For small numbers, it is generally feasible to use your knowledge of multiplication tables and factors to choose the largest common factor of two numbers. You can also use Euclid's algorithm, which is to repeatedly compute
... (largest number) modulo (smallest number)
until the result is zero. The final "smallest number" is the GCF.
Here, that looks like
... 72 mod 42 = 30
... 42 mod 30 = 12
... 30 mod 12 = 6
... 12 mod 6 = 0 . . . . . . . so 6 is the GCF
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Of course, you know that
... 72 = 2³×3²
... 42 = 2×3×7
so, the largest set of common factors is 2×3 = 6.
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Your graphing calculator may have a function for computing the greatest common divisor (GCD), too. The TI-84 does, for example.
Answer:
Step-by-step explanation:
If this is an exponential function, it is of the form
where x and y are coordinates from your table, a is the intial value, and b is the growth/decay rate. To find out what the equation is that represents this data, choose 2 points and solve first for a and then for b. Just a hint: If at all possible, choose the coordinate that gives you an x of 0. You'll see why in a minute.
I chose the first 2 points from your table: (0, .2) and (1, .8). Solving first for a:
The reason to choose the x of 0 as one of your points is because anything raised to the power of 0 is 1. So our equation then becomes:
so
a = .2 Easy enough.
Now use that value along with the other coordinate to solve for b:
b to the first is just b, so,
.8 = .2b
Divide both sides by .2 and you'll get that
b = 4
The equation, then, is
which is growth, since the value for b is greater than 1.