# Step-by-Step Solution: Angle by Repetition Given data: - **Initial reading:** \( 77^\circ\ 26'\ 30'' \) - **Final reading:** \( 104^\circ\ 38'\ 41'' \) (after **one full rotation**, so add \( 360^\circ \)) - **Number of repetitions:** \( 6 \) Let's find the value of the measured angle to the nearest second. --- ## 1. Adjust Final Reading The final reading was taken **after one full rotation** (i.e., \( 360^\circ \) added): \[ \text{Adjusted final reading} = 104^\circ\ 38'\ 41'' + 360^\circ = 464^\circ\ 38'\ 41'' \] --- ## 2. Find Total Angle Swept \[ \text{Total angle swept} = \text{Adjusted final reading} - \text{Initial reading} \] Write both values: - Initial: \( 77^\circ\ 26'\ 30'' \) - Final (adjusted): \( 464^\circ\ 38'\ 41'' \) Now, subtract: First, align the units for easy subtraction: \[ \begin{align*} 464^\circ\ 38'\ 41'' \\ - \underline{77^\circ\ 26'\ 30''} \\ \end{align*} \] Subtract seconds: \( 41'' - 30'' = 11'' \) Subtract minutes: \( 38' - 26' = 12' \) Subtract degrees: \( 464^\circ - 77^\circ = 387^\circ \) So, \[ \text{Total angle swept} = 387^\circ\ 12'\ 11'' \] --- ## 3. Calculate the Single Angle Number of repetitions = 6 \[ \text{Measured angle} = \frac{\text{Total angle swept}}{\text{Number of repetitions}} \] So, \[ \text{Measured angle} = \frac{387^\circ\ 12'\ 11''}{6} \] Let's divide each unit: ### a. Convert to Seconds Convert entire angle to seconds: - \( 1^\circ = 360'' \) - \( 1' = 60'' \) \[ 387^\circ\ 12'\ 11'' = (387 \times 360) + (12 \times 60) + 11 \] \[ 387 \times 360 = 1,\!393,\!200 \] \[ 12 \times 60 = 720 \] \[ \text{Total seconds} = 1,\!393,\!200 + 720 + 11 = 1,\!393,\!931'' \] ### b. Divide by 6 \[ \frac{1,\!393,\!931}{6} = 232,\!321.8333'' \] --- ### c. Convert Back to DMS - \( 232,\!321.8333'' \div 360 = 64.53384^\circ \) - So, degrees = 64 Remainder in seconds: \[ 232,\!321.8333'' - (64 \times 360) = 232,\!321.8333 - 230,400 = 1,921.8333'' \] Minutes: \[ 1,921.8333'' \div 60 = 32.03056' \] So, minutes = 32 Remainder in seconds: \[ 1,921.8333 - (32 \times 60) = 1,921.8333 - 1,920 = 1.8333'' \] --- ## 4. Final Answer (to the Nearest Second) \[ \boxed{64^\circ\ 32'\ 2''} \] --- ## **Summary Table** | Step | Value | |-----------|--------------------------------------| | Initial | \(77^\circ\ 26'\ 30''\) | | Final | \(104^\circ\ 38'\ 41''\) | | + 360° | \(464^\circ\ 38'\ 41''\) | | Difference| \(387^\circ\ 12'\ 11''\) | | / 6 | \(64^\circ\ 32'\ 2''\) (rounded) | --- ## **Final Answer** **The value of the angle to the nearest second is:** \[ \boxed{64^\circ\ 32'\ 2''} \]