Given:
\[ 3 = K(a - \ell) \] \[ 2 = K(b - \ell) \]
where \( K \) is the spring constant and \( \ell \) is the natural length of the spring.
To find the tension \( T \) for the length \( (3a - 2b) \), we use:
\[ T = K (3a - 2b - \ell) \]
Substituting the values of \( a - \ell \) and \( b - \ell \) from the given equations:
\[ T = K [3(a - \ell) - 2(b - \ell)] \]
\[ T = K \left[ 3 \left( \frac{3}{K} \right) - 2 \left( \frac{2}{K} \right) \right] \]
\[ T = K \left[ \frac{9}{K} - \frac{4}{K} \right] \]
\[ T = K \left[ \frac{5}{K} \right] = 5 \, \text{N} \]
Conclusion:
Hence, the value of the tension is \( 5 \, \text{N} \).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32