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Lena [83]
2 years ago
8

you make a necklace using blue, purple and green beads in a ratio of 1:1:2. you use a total of 168 beads. How many green beads a

re in the necklace?
Mathematics
2 answers:
Svetlanka [38]2 years ago
5 0

Answer:

84

Step-by-step explanation:

Given,

The ratio of blue, purple and green beads in the necklace = 1 : 1 : 2

Let blue beads = x, purple beads = x and green beads = 2x

Where,

x is any positive real number,

So, the total number of beads = x + x + 2x = 4x

According to the question,

4x = 168

x = 42,

Hence, the number of green beads in necklace = 2(42) = 84

vesna_86 [32]2 years ago
3 0
The answer is 84 green beads.

To get that you need to do 168 divide by 4 (four because I look at it like this 1:1:2 is like 1:1:1:1) That answer is 4 then double that and you get 84.
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Find a solution of x dy dx = y2 − y that passes through the indicated points. (a) (0, 1) y = (b) (0, 0) y = (c) 1 6 , 1 6 y = (d
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(a) y = \frac{1}{1 - Cx}, for any constant C

(b) Solution does not exist

(c) y = \frac{256}{256 - 15x}

(d) y = \frac{64}{64 - 15x}

Explanations:

(a) To solve the differential equation in the problem, we need to manipulate the equation such that the expression that involves y is on the left side of the equation and the expression that involves x is on the right side equation.

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 x\frac{dy}{dx} = y^2 - y&#10;\\&#10;\\ \indent xdy = \left ( y^2 - y \right )dx&#10;\\&#10;\\ \indent \frac{dy}{y^2 - y} = \frac{dx}{x}&#10;\\&#10;\\ \indent \int {\frac{dy}{y^2 - y}} = \int {\frac{dx}{x}} &#10;\\&#10;\\ \indent \boxed{\int {\frac{dy}{y^2 - y}} = \ln x + C_1}      (1)

Now, we need to evaluate the indefinite integral on the left side of equation (1). Note that the denominator y² - y = y(y - 1). So, the denominator can be written as product of two polynomials. In this case, we can solve the indefinite integral using partial fractions.

Using partial fractions:

\frac{1}{y^2 - y} = \frac{1}{y(y - 1)} = \frac{A}{y - 1} + \frac{B}{y}&#10;\\&#10;\\ \indent \Rightarrow \frac{1}{y^2 - y} = \frac{Ay + B(y-1)}{y(y - 1)} &#10;\\&#10;\\ \indent \Rightarrow \boxed{\frac{1}{y^2 - y} = \frac{(A+B)y - B}{y^2 - y} }      (2)

Since equation (2) has the same denominator, the numerator has to be equal. So,

1 = (A+B)y - B&#10;\\&#10;\\ \indent \Rightarrow (A+B)y - B = 0y + 1&#10;\\&#10;\\ \indent \Rightarrow \begin{cases}&#10; A + B = 0&#10;& \text{(3)}\\-B = 1&#10; & \text{(4)}   \end{cases}

Based on equation (4), B = -1. By replacing this value to equation (3), we have

A + B = 0
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Hence, 

\frac{1}{y^2 - y} = \frac{1}{y - 1} - \frac{1}{y}

So,

\int {\frac{dy}{y^2 - y}} = \int {\frac{dy}{y - 1}} - \int {\frac{dy}{y}} &#10;\\&#10;\\ \indent \indent \indent \indent = \ln (y-1) - \ln y&#10;\\&#10;\\ \indent  \boxed{\int {\frac{dy}{y^2 - y}} = \ln \left ( \frac{y-1}{y} \right ) + C_2}

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\ln \left ( \frac{y-1}{y} \right ) + C_2 = \ln x + C_1&#10;\\&#10;\\ \indent \ln \left ( \frac{y-1}{y} \right ) = \ln x + C_1 - C_2&#10;\\&#10;\\ \indent  \frac{y-1}{y} = e^{C_1 - C_2}x&#10;\\&#10;\\ \indent  \frac{y-1}{y} = Cx, \text{ where } C = e^{C_1 - C_2}&#10;\\&#10;\\ \indent  1 - \frac{1}{y} = Cx&#10;\\&#10;\\ \indent \frac{1}{y} = 1 - Cx&#10;\\&#10;\\ \indent \boxed{y = \frac{1}{1 - Cx}}&#10;       (5)

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\boxed{y = \frac{1}{1 - Cx}}

(b) Using equation (5) in problem (a),

y = \frac{1}{1 - Cx}   (6)

for any constant C.

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Note that 0 = 1 is false. Hence, for any constant C, the solution that passes thru (0,0) does not exist.

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y = \frac{1}{1 - Cx} &#10;\\ &#10;\\ \indent 16 = \frac{1}{1 - C(4)} &#10;\\ &#10;\\ \indent 16 = \frac{1}{1 - 4C} &#10;\\ &#10;\\ \indent 16(1 - 4C) = 1 &#10;\\ \indent 16 - 64C = 1 &#10;\\ \indent - 64C = -15 &#10;\\ \indent \boxed{C = \frac{15}{64}}

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