If \(\frac{2 a}{3}=\frac{4 b}{5}=\frac{3 c}{4}\), then what is the value of \(\frac{18}{a} \sqrt{a^2+c^2-b^2}\)?
Finding the Relationship between Variables
The problem provides a relationship between three variables: \(a\), \(b\), and \(c\). We are given:
\( \frac{2 a}{3}=\frac{4 b}{5}=\frac{3 c}{4} \)
To make calculations easier, let's set this common ratio equal to a constant, say \(k\).
\( \frac{2 a}{3} = k \quad \implies \quad a = \frac{3k}{2} \)
\( \frac{4 b}{5} = k \quad \implies \quad b = \frac{5k}{4} \)
\( \frac{3 c}{4} = k \quad \implies \quad c = \frac{4k}{3} \)
Now we have expressions for \(a\), \(b\), and \(c\) in terms of \(k\). This allows us to find the value of the required expression.
Calculating Expression Components
The expression we need to evaluate is \(\frac{18}{a} \sqrt{a^2+c^2-b^2}\). We need to calculate the terms \(a^2\), \(c^2\), \(b^2\), and then the term inside the square root, \(a^2+c^2-b^2\). We also need the term \(\frac{18}{a}\).
Calculating Squared Variables
Using the expressions for \(a\), \(b\), and \(c\) derived above:
\( a^2 = \left(\frac{3k}{2}\right)^2 = \frac{9k^2}{4} \)
\( b^2 = \left(\frac{5k}{4}\right)^2 = \frac{25k^2}{16} \)
\( c^2 = \left(\frac{4k}{3}\right)^2 = \frac{16k^2}{9} \)
Simplifying the Term Under the Square Root
Now, substitute these squared values into the expression \(a^2+c^2-b^2\):
\( a^2+c^2-b^2 = \frac{9k^2}{4} + \frac{16k^2}{9} - \frac{25k^2}{16} \)
To combine these fractions, we find a common denominator for 4, 9, and 16. The least common multiple (LCM) of 4, 9, and 16 is 144.
\( a^2+c^2-b^2 = k^2 \left( \frac{9}{4} + \frac{16}{9} - \frac{25}{16} \right) \)
\( a^2+c^2-b^2 = k^2 \left( \frac{9 \times 36}{144} + \frac{16 \times 16}{144} - \frac{25 \times 9}{144} \right) \)
\( a^2+c^2-b^2 = k^2 \left( \frac{324}{144} + \frac{256}{144} - \frac{225}{144} \right) \)
\( a^2+c^2-b^2 = k^2 \left( \frac{324 + 256 - 225}{144} \right) \)
\( a^2+c^2-b^2 = k^2 \left( \frac{580 - 225}{144} \right) \)
\( a^2+c^2-b^2 = k^2 \left( \frac{355}{144} \right) \)
Now, we take the square root of this result:
\( \sqrt{a^2+c^2-b^2} = \sqrt{k^2 \left( \frac{355}{144} \right)} \)
\( \sqrt{a^2+c^2-b^2} = |k| \frac{\sqrt{355}}{\sqrt{144}} \)
Assuming \(k\) is positive, we get:
\( \sqrt{a^2+c^2-b^2} = k \frac{\sqrt{355}}{12} \)
Evaluating the Final Expression
Next, let's calculate the term \(\frac{18}{a}\):
\( \frac{18}{a} = \frac{18}{\frac{3k}{2}} \)
\( \frac{18}{a} = 18 \times \frac{2}{3k} \)
\( \frac{18}{a} = \frac{36}{3k} \)
\( \frac{18}{a} = \frac{12}{k} \)
Finally, we multiply the two parts together:
\( \frac{18}{a} \sqrt{a^2+c^2-b^2} = \left( \frac{12}{k} \right) \times \left( k \frac{\sqrt{355}}{12} \right) \)
The \(k\) terms cancel out, and the 12s also cancel out:
\( \frac{18}{a} \sqrt{a^2+c^2-b^2} = \sqrt{355} \)
Thus, the value of the expression is \(\sqrt{355}\).
Similar Questions
Which one of the following fractions will have minimum change in its value if 3 is added to both the numerator and the denominator of all the fractions?
A 2-digit number is such that the sum of the number and the number obtained by reversing the order of the digits of the number is 55. Further, the difference of the given number and the number obtained by reversing the order of the digits of the number is 45. What is the product of the digits?
The cube root of x varies inversely as the square root of y. x = 8 when y = 3. What is the value of x when y = \(\sqrt[3]{3} \) ?
Important Questions from Ratio and Proportion
In a mixture of 156 litres, the ratio of milk and water is 7 : 6. How much water should be added to make the ratio 14 : 13?
If the ratio of the first to second number is 3 : 4 and that of the second to the third number is 8 : 5, and sum of three numbers is 190 then the third number is:
The third proportional to 9 and 15 is:
The average age of 3 persons is 30 years.If their ages are in the ratio of 3 : 5 : 7 respectively, then the age of the eldest person is:
The income of A and B are in the ratio 5 : 3. The expenses of A, B and C are in the ratio of 8 : 5 : 2. If C spends 2000 and B saves ₹ 700, then A saves:
