At a mountain expedition camp, Emek and Tara received an emergency message from hikers stranded beyond two possible rescue routes. For Route A, changing terrain made the travel time t satisfy 6t + 12 = 42, so Tara isolated t and found a travel time of 5 hours. Route B was 24 kilometers long, but the team's speed depended on the trip time because of a steep climb. Its model was 24 = t(10 - t). Emek rearranged it to t^2 - 10t + 24 = 0 and factored the equation as (t - 4)(t - 6) = 0. The two mathematical solutions were 4 and 6 hours, but the route report stated that safe conditions required a speed of at least 5 kilometers per hour. At t = 6, the model gave only 4 kilometers per hour, so that solution was invalid. The valid Route B time was 4 hours. Emek and Tara recommended Route B, saving an hour and helping the rescue team reach the hikers safely.

Challenges

Start

Challenge 1 of 5

Challenges

Start

Challenge 1 of 4

Hint

Undo the addition of 12 first, then divide by 6.

For Route A, which sequence correctly solves 6t + 12 = 42?

6t = 54, so t = 9 6t = 30, so t = 5 6t = 42, so t = 7 6t = 36, so t = 6

Correct. Subtracting 12 gives 6t = 30, so t = 5 hours.

Isolate the term containing t by subtracting 12 from both sides first.

🎉 Congrats! You got this question!

Hint

Replace t by 4 in both places and evaluate the expression.

Emek checks the claimed Route B solution t = 4 by substituting it into the model t(10 - t). What integer value does the expression t(10 - t) equal when t = 4?

Correct. Substituting 4 gives 4(10 - 4) = 24.

Evaluate 10 - 4 first, then multiply the result by 4.

🎉 Congrats! You got this question!

Hint

Substitute each proposed value directly into the original Route B equation.

The story states that t = 4 and t = 6 solve the Route B equation 24 = t(10 - t). Which statement correctly checks this claim?

Both values are solutions because 4 + 6 = 10.Only t = 4 is a solution because 4(10 - 4) = 36.Only t = 6 is a solution because 6(10 - 6) = 36.Both values are solutions because both 4(10 - 4) and 6(10 - 6) equal 24.

Correct. Direct substitution shows that both 4 and 6 satisfy the stated equation.

Check each proposed solution in the original equation rather than relying only on the stated factorization.

🎉🎉🎉🎉🎉🎉🎉🎉

🎉 Congrats! You finished this challenge set!

Dağ Kurtarma Rotası