Solved on Jan 15, 2024

Find the values of $a$ , $b$ , and $c$ given the system of proportions $\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}$ .

STEP 1

Assumptions
1. The given equation is $\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}$ .
2. We need to find the values of $a$ , $b$ , and $c$ .
3. All variables are non-zero to prevent division by zero.

STEP 2

Since the four fractions are equal, we can set up a system of equations by equating each pair of fractions.
$\frac{5}{a} = \frac{b}{27}$ $\frac{b}{27} = \frac{12 \sqrt{3}}{c}$ $\frac{12 \sqrt{3}}{c} = \frac{9}{b}$

STEP 3

Solve the first equation for $b$ .
$b = \frac{27 \cdot 5}{a}$

STEP 4

Substitute the expression for $b$ from STEP_3 into the second equation.
$\frac{27 \cdot 5}{a \cdot 27} = \frac{12 \sqrt{3}}{c}$

STEP 5

Simplify the left side of the equation from STEP_4.
$\frac{5}{a} = \frac{12 \sqrt{3}}{c}$

STEP 6

Now, we can equate the expression from STEP_5 with the third equation.
$\frac{5}{a} = \frac{9}{b}$

STEP 7

Substitute the expression for $b$ from STEP_3 into the equation from STEP_6.
$\frac{5}{a} = \frac{9}{\frac{27 \cdot 5}{a}}$

STEP 8

Solve the equation from STEP_7 for $a$ .
$a^2 = \frac{27 \cdot 5^2}{9}$

STEP 9

Simplify the right side of the equation from STEP_8.
$a^2 = \frac{27 \cdot 25}{9}$

STEP 10

Further simplify the right side of the equation from STEP_9.
$a^2 = 3 \cdot 25$

STEP 11

Take the square root of both sides of the equation from STEP_10 to solve for $a$ .
$a = 5 \sqrt{3}$

STEP 12

Now that we have $a$ , we can find $b$ using the expression from STEP_3.
$b = \frac{27 \cdot 5}{5 \sqrt{3}}$

STEP 13

Simplify the expression for $b$ from STEP_12.
$b = \frac{27}{\sqrt{3}}$

STEP 14

Rationalize the denominator of the expression for $b$ from STEP_13.
$b = 27 \cdot \frac{\sqrt{3}}{\sqrt{3}}$

STEP 15

Simplify the expression for $b$ from STEP_14.
$b = 9 \sqrt{3}$

STEP 16

Now we can find $c$ using the second equation.
$\frac{9 \sqrt{3}}{27} = \frac{12 \sqrt{3}}{c}$

STEP 17

Simplify the left side of the equation from STEP_16.
$\frac{1}{3} = \frac{12 \sqrt{3}}{c}$

STEP 18

Solve the equation from STEP_17 for $c$ .
$c = 12 \sqrt{3} \cdot 3$

STEP 19

Simplify the expression for $c$ from STEP_18.
$c = 36 \sqrt{3}$
The values of $a$ , $b$ , and $c$ are $5 \sqrt{3}$ , $9 \sqrt{3}$ , and $36 \sqrt{3}$ respectively.

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Find the values of aaa, bbb, and ccc given the system of proportions 5a=b27=123c=9b\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}a5​=27b​=c123​​=b9​.

STEP 1

Assumptions1. The given equation is 5a=b27=123c=9b\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}a5​=27b​=c123​​=b9​.2. We need to find the values of aaa, bbb, and ccc.3. All variables are non-zero to prevent division by zero.

STEP 2

Since the four fractions are equal, we can set up a system of equations by equating each pair of fractions.5a=b27\frac{5}{a} = \frac{b}{27}a5​=27b​ b27=123c\frac{b}{27} = \frac{12 \sqrt{3}}{c}27b​=c123​​ 123c=9b\frac{12 \sqrt{3}}{c} = \frac{9}{b}c123​​=b9​

STEP 3

Solve the first equation for bbb.b=27⋅5ab = \frac{27 \cdot 5}{a}b=a27⋅5​

STEP 4

Substitute the expression for bbb from STEP_3 into the second equation.27⋅5a⋅27=123c\frac{27 \cdot 5}{a \cdot 27} = \frac{12 \sqrt{3}}{c}a⋅2727⋅5​=c123​​

STEP 5

Simplify the left side of the equation from STEP_4.5a=123c\frac{5}{a} = \frac{12 \sqrt{3}}{c}a5​=c123​​

STEP 6

Now, we can equate the expression from STEP_5 with the third equation.5a=9b\frac{5}{a} = \frac{9}{b}a5​=b9​

STEP 7

Substitute the expression for bbb from STEP_3 into the equation from STEP_6.5a=927⋅5a\frac{5}{a} = \frac{9}{\frac{27 \cdot 5}{a}}a5​=a27⋅5​9​

STEP 8

Solve the equation from STEP_7 for aaa.a2=27⋅529a^2 = \frac{27 \cdot 5^2}{9}a2=927⋅52​

STEP 9

Simplify the right side of the equation from STEP_8.a2=27⋅259a^2 = \frac{27 \cdot 25}{9}a2=927⋅25​

STEP 10

Further simplify the right side of the equation from STEP_9.a2=3⋅25a^2 = 3 \cdot 25a2=3⋅25

STEP 11

Take the square root of both sides of the equation from STEP_10 to solve for aaa.a=53a = 5 \sqrt{3}a=53​

STEP 12

Now that we have aaa, we can find bbb using the expression from STEP_3.b=27⋅553b = \frac{27 \cdot 5}{5 \sqrt{3}}b=53​27⋅5​

STEP 13

Simplify the expression for bbb from STEP_12.b=273b = \frac{27}{\sqrt{3}}b=3​27​

STEP 14

Rationalize the denominator of the expression for bbb from STEP_13.b=27⋅33b = 27 \cdot \frac{\sqrt{3}}{\sqrt{3}}b=27⋅3​3​​

STEP 15

Simplify the expression for bbb from STEP_14.b=93b = 9 \sqrt{3}b=93​

STEP 16

Now we can find ccc using the second equation.9327=123c\frac{9 \sqrt{3}}{27} = \frac{12 \sqrt{3}}{c}2793​​=c123​​

STEP 17

Simplify the left side of the equation from STEP_16.13=123c\frac{1}{3} = \frac{12 \sqrt{3}}{c}31​=c123​​

STEP 18

Solve the equation from STEP_17 for ccc.c=123⋅3c = 12 \sqrt{3} \cdot 3c=123​⋅3

STEP 19

Simplify the expression for ccc from STEP_18.c=363c = 36 \sqrt{3}c=363​The values of aaa, bbb, and ccc are 535 \sqrt{3}53​, 939 \sqrt{3}93​, and 36336 \sqrt{3}363​ respectively.

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Find the values of $a$ , $b$ , and $c$ given the system of proportions $\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}$ .

Assumptions
1. The given equation is $\frac{5}{a}=\frac{b}{27}=\frac{12 \sqrt{3}}{c}=\frac{9}{b}$ .
2. We need to find the values of $a$ , $b$ , and $c$ .
3. All variables are non-zero to prevent division by zero.

Since the four fractions are equal, we can set up a system of equations by equating each pair of fractions.
$\frac{5}{a} = \frac{b}{27}$ $\frac{b}{27} = \frac{12 \sqrt{3}}{c}$ $\frac{12 \sqrt{3}}{c} = \frac{9}{b}$

Solve the first equation for $b$ .
$b = \frac{27 \cdot 5}{a}$

Substitute the expression for $b$ from STEP_3 into the second equation.
$\frac{27 \cdot 5}{a \cdot 27} = \frac{12 \sqrt{3}}{c}$

Simplify the left side of the equation from STEP_4.
$\frac{5}{a} = \frac{12 \sqrt{3}}{c}$

Now, we can equate the expression from STEP_5 with the third equation.
$\frac{5}{a} = \frac{9}{b}$

Substitute the expression for $b$ from STEP_3 into the equation from STEP_6.
$\frac{5}{a} = \frac{9}{\frac{27 \cdot 5}{a}}$

Solve the equation from STEP_7 for $a$ .
$a^2 = \frac{27 \cdot 5^2}{9}$

Simplify the right side of the equation from STEP_8.
$a^2 = \frac{27 \cdot 25}{9}$

Further simplify the right side of the equation from STEP_9.
$a^2 = 3 \cdot 25$

Take the square root of both sides of the equation from STEP_10 to solve for $a$ .
$a = 5 \sqrt{3}$

Now that we have $a$ , we can find $b$ using the expression from STEP_3.
$b = \frac{27 \cdot 5}{5 \sqrt{3}}$

Simplify the expression for $b$ from STEP_12.
$b = \frac{27}{\sqrt{3}}$

Rationalize the denominator of the expression for $b$ from STEP_13.
$b = 27 \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Simplify the expression for $b$ from STEP_14.
$b = 9 \sqrt{3}$

Now we can find $c$ using the second equation.
$\frac{9 \sqrt{3}}{27} = \frac{12 \sqrt{3}}{c}$

Simplify the left side of the equation from STEP_16.
$\frac{1}{3} = \frac{12 \sqrt{3}}{c}$

Solve the equation from STEP_17 for $c$ .
$c = 12 \sqrt{3} \cdot 3$

Simplify the expression for $c$ from STEP_18.
$c = 36 \sqrt{3}$
The values of $a$ , $b$ , and $c$ are $5 \sqrt{3}$ , $9 \sqrt{3}$ , and $36 \sqrt{3}$ respectively.