Questions: The local police department gathered speeds of drivers on a road with a speed limit of 35. They tabulated the speeds of drivers over a 24 hour period, then averaged those speeds by the time of day. Results are provided in the table below. Calculate the average speed driven on this road. Round your answer to the nearest tenth. Time of Day Percent of Drivers Speed (in mph) 12am-5:59am 9% 39.9 6am-11:59am 26% 32.4 12 pm-5: 59 pm 41% 34.6 6 pm-11: 59 pm 24% 44.3

Solution Steps

To calculate the average speed driven on the road, we need to find the weighted average of the speeds, where the weights are the percentages of drivers for each time period. Multiply each speed by its corresponding percentage (converted to a decimal), sum these products, and then divide by the total percentage (which should be 100%).

To calculate the weighted average, we first convert the percentages of drivers into decimal form. This is done by dividing each percentage by 100: \[ \text{Weights} = \left[ \frac{9}{100}, \frac{26}{100}, \frac{41}{100}, \frac{24}{100} \right] = [0.09, 0.26, 0.41, 0.24] \]

Next, we calculate the weighted average speed by multiplying each speed by its corresponding weight and summing the results: \[ \text{Average Speed} = (0.09 \times 39.9) + (0.26 \times 32.4) + (0.41 \times 34.6) + (0.24 \times 44.3) \] \[ = 3.591 + 8.424 + 14.186 + 10.632 = 36.833 \]

Finally, we round the average speed to the nearest tenth: \[ \text{Average Speed Rounded} = 36.8 \]

Final Answer