Karen drove at an average speed of \( 45 \mathrm{mi} / \mathrm{h} \) from her home in Cincinnati to visit her sister in Atlanta. She stayed in Atlanta 10 hours, and on the trip back averaged \( 45 \mathrm{mi} / \mathrm{h} \). She returned home 44 hours after leaving. How many miles is Cincinnati from Atlanta? a) Write an equation using the information as it is given above that can be solved to answer this question. Use \( t \) as your variable to represent the amount of time Karen spent driving from Cincinnati to Atlanta. Equation: b) How many miles is Cincinnati from Atlanta? Answer:

Let's start by crafting the equation that captures the situation. a) Karen's total time for the round trip includes the time spent driving to Atlanta, the 10 hours she stayed there, and the time spent driving back home. Since she drove at an average speed of 45 miles per hour (mi/h) both ways, we can express the time taken to drive from Cincinnati to Atlanta as \( t \) hours, which means the distance \( d \) from Cincinnati to Atlanta is \( 45t \) miles. The return trip, being the same distance, also takes \( t \) hours. Thus, the total equation combining all parts of her trip becomes: \[ t + 10 + t = 44 \] This simplifies to: \[ 2t + 10 = 44 \] b) We can solve for \( t \) from the equation: \[ 2t + 10 = 44 \] \[ 2t = 44 - 10 \] \[ 2t = 34 \] \[ t = 17 \text{ hours} \] Now that we know it took her 17 hours to drive to Atlanta, we can find the distance: \[ d = 45t = 45 \times 17 = 765 \text{ miles} \] So, the distance from Cincinnati to Atlanta is **765 miles**.