14. Two cars leave town at the same time traveling in opposite directions. One travels \( 51 \mathrm{mi} / \mathrm{h} \) and the other travels \( 45 \mathrm{mi} / \mathrm{h} \). In how many hours will they be 432 miles apart?

To find out how many hours it will take for the two cars to be 432 miles apart, we can use the concept of relative speed. Let's denote the speed of the first car as \( v_1 = 51 \, \text{mi/h} \) and the speed of the second car as \( v_2 = 45 \, \text{mi/h} \). The relative speed of the two cars is the sum of their individual speeds because they are moving in opposite directions. Therefore, the relative speed is \( v_1 + v_2 = 51 \, \text{mi/h} + 45 \, \text{mi/h} = 96 \, \text{mi/h} \). Now, we can use the formula for distance, which is \( \text{distance} = \text{speed} \times \text{time} \), to find the time it will take for the two cars to be 432 miles apart. Let's denote the time as \( t \) hours. The distance covered by the first car in time \( t \) is \( v_1 \times t \) miles, and the distance covered by the second car in time \( t \) is \( v_2 \times t \) miles. Since they are moving in opposite directions, the total distance between them is the sum of the distances covered by each car, which is \( v_1 \times t + v_2 \times t = 432 \) miles. Therefore, we have the equation \( 51t + 45t = 432 \). Solving this equation will give us the time it will take for the two cars to be 432 miles apart. Solve the equation by following steps: - step0: Solve for \(t\): \(51t+45t=432\) - step1: Add the terms: \(96t=432\) - step2: Divide both sides: \(\frac{96t}{96}=\frac{432}{96}\) - step3: Divide the numbers: \(t=\frac{9}{2}\) The time it will take for the two cars to be 432 miles apart is \( t = 4.5 \) hours.