ind the length of the line given by \( r(t)=\langle 9 t, 10 t\rangle \) for \( \mathrm{a} \leq t \leq \mathrm{b} \). he length of the line is Type an expression using a and b as the variables. Type an exact answer, using radicals as neede

To find the length of the line given by the vector function \( r(t) = \langle 9t, 10t \rangle \) for \( a \leq t \leq b \), we start by calculating the derivative \( r'(t) \): \[ r'(t) = \langle 9, 10 \rangle. \] Next, we find the magnitude of this derivative: \[ \| r'(t) \| = \sqrt{9^2 + 10^2} = \sqrt{81 + 100} = \sqrt{181}. \] The length \( L \) of the line segment as \( t \) goes from \( a \) to \( b \) is given by: \[ L = \int_{a}^{b} \| r'(t) \| \, dt = \int_{a}^{b} \sqrt{181} \, dt = \sqrt{181} \cdot (b - a). \] Therefore, the expression for the length of the line segment is: \[ L = \sqrt{181} (b - a). \]