During a canyon expedition, Emek and Tara needed to choose a safe location for a temporary bridge. Their mapping system estimated the required cable length, in meters, with the expression 3(x + 8) + 2(x - 4), where x represented the canyon width. Emek expanded the expression to get 3x + 24 + 2x - 8, then combined like terms to obtain 5x + 16. The team was considering two stable crossing points: Point A was 18 meters wide, and Point B was 23 meters wide. Substitution gave cable requirements of 106 meters at Point A and 131 meters at Point B. Tara then rewrote 5x + 16 as 5(x + 3.2), showing that the cable requirement increased directly with the adjusted width x + 3.2. The factored form also made clear how strongly changes in canyon width affected the total. Since both locations were safe but Point A required 25 meters less cable, Emek and Tara chose Point A and began preparing the bridge.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
Distribute each coefficient to both terms inside its parentheses before combining like terms.
Which sequence correctly simplifies the cable-length expression 3(x + 8) + 2(x - 4)?
3x + 8 + 2x - 4 = 5x + 43x + 24 + 2x - 8 = 5x + 163x + 24 + 2x + 8 = 5x + 326x + 24 + 2x - 4 = 8x + 20
Correct. Distributing and combining like terms gives 5x + 16.
Not quite. Check that 3 and 2 are each distributed to both terms in their parentheses.
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Hint
First find the new canyon width, then substitute it for x in 5x + 16.
The simplified cable requirement is 5x + 16 meters. The team finds another safe crossing point that is 4 meters wider than Point A, which is 18 meters wide. How many meters of cable would this new crossing point require?
Correct. The new crossing point would require 126 meters of cable.
Check the new width first, then evaluate 5x + 16 using that value.
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Hint
In 5x + 16, identify how much the output changes when x increases by 1 meter.
Point B is 5 meters wider than Point A. Without separately calculating the full cable requirement at each point, how can the simplified expression 5x + 16 be used to determine the difference in cable needed?
Multiply the 5-meter increase in width by the coefficient 5, giving 25 meters more cable.Add the constant 16 to the 5-meter increase, giving 21 meters more cable.Multiply the 5-meter increase by 16, giving 80 meters more cable.Add 5 and 16, then multiply by 5, giving 105 meters more cable.
Correct. Each additional meter of canyon width adds 5 meters of cable, so 5 extra meters of width add 25 meters of cable.
Not quite. Focus on the coefficient of x, which tells how the cable requirement changes as the canyon width changes.
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Kanyonu Geçmek
TR
ARC_G11_OPS_02_STORY_01
Kanyonu Geçmek
Crossing the Canyon
EN
ARC_G11_OPS_02_STORY_01
Crossing the Canyon