Approximate the integral below using a right hand sum with 5 rectangles. Round your final answer to the nearest thousandth. \[ \int_{4}^{9}\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right) d x \approx[?] \]

To approximate the integral \(\int_{4}^{9}\left(\frac{1-\sqrt{x}}{\sqrt{x}}\right) d x\) using a right-hand Riemann sum with 5 rectangles, we first calculate the width of each rectangle. The interval \([4, 9]\) has a width of \(9 - 4 = 5\). Dividing this into 5 equal parts gives us a width of: \[ \Delta x = \frac{5}{5} = 1 \] Next, the right endpoints for the rectangles are \(x = 5, 6, 7, 8, 9\). Now we evaluate the function \(\frac{1-\sqrt{x}}{\sqrt{x}}\) at each of these points: 1. At \(x = 5\): \[ f(5) = \frac{1 - \sqrt{5}}{\sqrt{5}} \approx \frac{1 - 2.236}{2.236} \approx \frac{-1.236}{2.236} \approx -0.553 \] 2. At \(x = 6\): \[ f(6) = \frac{1 - \sqrt{6}}{\sqrt{6}} \approx \frac{1 - 2.449}{2.449} \approx \frac{-1.449}{2.449} \approx -0.592 \] 3. At \(x = 7\): \[ f(7) = \frac{1 - \sqrt{7}}{\sqrt{7}} \approx \frac{1 - 2.646}{2.646} \approx \frac{-1.646}{2.646} \approx -0.622 \] 4. At \(x = 8\): \[ f(8) = \frac{1 - \sqrt{8}}{\sqrt{8}} \approx \frac{1 - 2.828}{2.828} \approx \frac{-1.828}{2.828} \approx -0.646 \] 5. At \(x = 9\): \[ f(9) = \frac{1 - \sqrt{9}}{\sqrt{9}} = \frac{1 - 3}{3} = \frac{-2}{3} \approx -0.667 \] Now we can compute the Riemann sum: \[ \text{Riemann Sum} = \Delta x \cdot \sum_{i=1}^{5} f(x_i) = 1 \cdot (f(5) + f(6) + f(7) + f(8) + f(9)) \] Calculating the sum: \[ f(5) + f(6) + f(7) + f(8) + f(9) \approx -0.553 - 0.592 - 0.622 - 0.646 - 0.667 \approx -3.080 \] So, the Riemann sum is: \[ 1 \cdot (-3.080) = -3.080 \] Rounding to the nearest thousandth, the final answer is \(\approx -3.080\).