Blue Glacier Supplies makes rock climbing ropes in different lengths. The factory has 2,500 meters of rope that will be cut to make some shorter 60-meter ropes and some longer 70-meter ropes. The factory owner wants to produce at least 40 ropes.
Which pairs (s,𝓁) represent possible numbers of shorter and longer ropes the factory could make?
a) (34, 6)
b) (45, -3)
c) (25,14)
d) (30,10)
The total length of rope available is 2500 meters. Let x be the number of 60-meter ropes and y be the number of 70-meter ropes produced.
We have the following equations:
60x + 70y = 2500 (total length of available rope)
x + y ≥ 40 (at least 40 ropes produced)
Now, let's check each pair (s, l) to see if they satisfy the above equations:
a) (34, 6)
60(34) + 70(6) = 2040 + 420 = 2460 (total length of available rope) - satisfies the first equation
34 + 6 = 40 - satisfies the second equation
Therefore, (34, 6) is a possible pair.
b) (45, -3)
Since we cannot have negative ropes, this pair is not valid.
c) (25, 14)
60(25) + 70(14) = 1500 + 980 = 2480 (total length of available rope) - satisfies the first equation
25 + 14 = 39 < 40 - does not satisfy the second equation
Therefore, (25, 14) is not a possible pair.
d) (30, 10)
60(30) + 70(10) = 1800 + 700 = 2500 (total length of available rope) - satisfies the first equation
30 + 10 = 40 - satisfies the second equation
Therefore, (30, 10) is a possible pair.
Therefore, the possible pairs are:
a) (34, 6)
d) (30, 10)