During a long space mission, Emek and Tara planned six course-correction maneuvers while protecting enough fuel for the journey home. The first maneuver required 160 kilograms of fuel, and each later maneuver was expected to use 75 percent of the amount used previously. Tara recognized a geometric sequence with first term 160 and common ratio 0.75. Using a_n = 160(0.75)^(n - 1), she found that the sixth maneuver would require 37.96875 kilograms. Emek then calculated the total fuel needed for all six maneuvers using the geometric series formula S_n = 160(1 - 0.75^n)/(1 - 0.75). The result was 525.15625 kilograms. Their spacecraft had 980 kilograms available before the corrections, but mission rules required at least 440 kilograms to remain for the return journey. After the six maneuvers, 454.84375 kilograms would remain. Since this exceeded the required reserve by 14.84375 kilograms, Emek and Tara approved the maneuver plan and continued the mission with enough fuel protected for the trip home.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
The question asks for the total of several terms, so use a geometric series rather than the formula for one term.
Emek wants to calculate the total fuel used in the first four maneuvers. Which expression correctly represents this total?
160(0.75)^3160(1 - 0.75^4)/(1 - 0.75)160(1 - 0.75^3)/(1 - 0.75)4(160)(0.75)
Correct. The sum of the first four terms uses the geometric series formula with n = 4.
Not quite. Distinguish between finding one maneuver's fuel amount and finding the total fuel for four maneuvers.
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Hint
Subtract the required reserve from the amount of fuel available before the maneuvers.
Mission rules require at least 440 kilograms of the original 980 kilograms to remain for the return journey. What is the greatest whole number of kilograms that can be used for course-correction maneuvers while still meeting this reserve requirement?
Correct. The spacecraft can use at most 540 kilograms and still keep the required 440 kilograms.
Check how much of the 980 kilograms can be used after protecting the 440-kilogram reserve.
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Hint
Multiply the sixth maneuver's fuel amount by 0.75, then add the result to the six-maneuver total and compare with the maximum usable fuel.
The six planned maneuvers use 525.15625 kilograms in total. If Emek and Tara add a seventh maneuver that follows the same geometric pattern, which conclusion is correct?
The seventh maneuver uses 37.96875 kilograms, so the total remains below the 540-kilogram fuel limit.The seventh maneuver uses 28.4765625 kilograms, so the new total is 553.6328125 kilograms and the reserve requirement is not met.The seventh maneuver uses 28.4765625 kilograms, so the new total is 496.6796875 kilograms and the reserve requirement is met.The seventh maneuver uses 21.357421875 kilograms, so the new total remains below the 540-kilogram fuel limit.
Correct. Adding a seventh maneuver would raise total fuel use above 540 kilograms, leaving less than the required reserve.
Not quite. First find the seventh term using the common ratio, then add it to the six-maneuver total and check the reserve.
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Yörünge Manevraları İçin Yakıt
TR
ARC_G12_PA_02_STORY_02
Yörünge Manevraları İçin Yakıt
Fuel for the Orbital Maneuvers
EN
ARC_G12_PA_02_STORY_02
Fuel for the Orbital Maneuvers