The Jacksons and the Simpsons were competing in the final leg of the Amazing Race, which was 240 kilometers long. In their race to the finish, the Jacksons immediately took off traveling at an average speed of v kilometers per hour. The Simpsons' start was delayed by an hour. When they eventually took off, they traveled at an average speed that was 40 kilometers per hour faster than the Jacksons' speed. Sadly for them, that didn't help, and the Jacksons won. Write an inequality in terms of v that models the situation.

Question

The Jacksons and the Simpsons were competing in the final leg of the Amazing Race, which was 240 kilometers long. In their race to the finish, the Jacksons immediately took off traveling at an average speed of  v kilometers per hour. The Simpsons' start was delayed by an hour. When they eventually took off, they traveled at an average speed that was 40 kilometers per hour faster than the Jacksons' speed. Sadly for them, that didn't help, and the Jacksons won. Write an inequality in terms of  v that models the situation.

Bytelearn · Accepted Answer

Equation for Time: Let's denote the time it takes for the Jacksons to complete the race as t hours. Since the Jacksons' speed is v kilometers per hour and they travel 240 kilometers, we can write the equation for the time it takes them to finish the race as: 240 = v * t Now, we solve for t: t = 240 / vInequality for Simpsons: The Simpsons start 1 hour later than the Jacksons, so the time available for them to catch up and potentially win the race is t - 1 hours. Their speed is v + 40 kilometers per hour. To find out if they can win or not, we need to set up an inequality that represents the distance they can travel in t - 1 hours at their speed, which should be less than or equal to 240 kilometers (the total race distance) for the Jacksons to win. The inequality will be: (v + 40) * (t - 1) ≤ 240Substitute Time in Inequality: We already know that t = 240 / v from the first step. Let's substitute this value into the inequality: (v + 40) * ((240 / v) - 1) ≤ 240 Now, we simplify the inequality: 240 + 40t - v - 40 ≤ 240Simplify Inequality: We substitute t with 240 / v in the term 40t: 240 + (40 * (240 / v)) - v - 40 ≤ 240 Now, we simplify the inequality further: 240 + 9600 / v - v - 40 ≤ 240Final Inequality: We can subtract 240 from both sides of the inequality to simplify it: 9600 / v - v ≤ 0 This is the inequality that models the situation. To ensure that the Jacksons win, the left side of the inequality must be less than or equal to zero.

Full solution

More problems from Write a linear inequality: word problems

Related topics