Final answer:
The mathematics question deals with two cyclists biking away from each other with one cyclist traveling at 12 mph and the second at "b" mph faster. After 45 minutes, they are 26.5 miles apart, leading to an equation that can be used to solve for "b". The equation is 9 + (12 + b) × 0.75 = 19.5.
Step-by-step explanation:
The problem involves two cyclists starting 7 miles apart and cycling away from each other. Cyclist one travels at a speed of 12 miles per hour, and cyclist two travels at a speed b miles per hour faster than cyclist one. After 45 minutes, they are 26.5 miles apart. To build an equation that represents this situation, we use the fact that distance is equal to speed multiplied by time.
The combined distance covered by both cyclists can be calculated by adding the distance each one travels. Since they start 7 miles apart, and after 45 minutes (which is 0.75 hours) they are 26.5 miles apart, the total distance covered by both cyclists during the 45 minutes is 26.5 miles - 7 miles.
Let's call the distance the first cyclist travels D1 and the distance the second cyclist travels D2. Thus, D1 + D2 = 19.5 miles. Since the first cyclist travels at 12 mph, we have D1 = 12 miles/hour × 0.75 hours = 9 miles. The second cyclist travels at (12 + b) mph, so D2 = (12 + b) miles/hour × 0.75 hours.
Substituting these distances into the combined distance, we get the equation 9 miles + (12 + b) × 0.75 = 19.5 miles. This is the equation needed to solve for b.