Answer:
Let a= number of adults and c= number of children. Then 150a is the total weight contributed by the adults, and 75c is the total weight contributed by the children. Since 10 pounds of gear is required per adult and per child, we need to add 10a and 10c to each of these amounts.
150a+10a=160a
75c+10c=85c
So the total weight in the boat contributed solely by the people is
160a+85c
Because each group also requires 200 pounds of gear regardless of how many people there are, we add 200 to the above amount. We also know that the total weight cannot exceed 1200 pounds. So, we arrive at the following inequality:
160a+85c+200≤1200
or
160a+85c≤1000
To write an inequality for the passenger limit in a boat, we observe that the total number of people aboard is the number of adults, a, added to the number of children, c. Since the number aboard cannot exceed 8, we arrive at
a+c≤8
We now have a system of linear inequalities:
160a+85c≤1000
a+c≤8.
The graph of the two inequalities is shown below. Note that any solution corresponds to a coordinate point (c,a) that lies in the doubly shaded region and where both coordinates are non-negative integers.
Fishing3_ff90c4880725b4d33d0174d24242fb13
We can find out which of the groups, if any, can safely rent a boat by first noting that all groups have less than 8 total people, so the passenger limit inequality is satisfied. Substituting the number of adults and children in each group for a and c in our weight inequality, we see that
For Group 1:160(4)+85(2)+200=1010≤1200
For Group 2:160(3)+85(5)+200=1105≤1200
For Group 3:160(8)+200=1480≰1200
We find that both Group 1 and Group 2 can safely rent a boat, but that Group 3 exceeds the weight limit, and so cannot rent a boat.
We could also have done a visual check to see which of the point (2,4), (5,3), and (0,8) lies in the doubly shaded region.
Other combinations of adults and children are possible, and can be found easily by looking at our graph. Any combination where (c,a) lies in the doubly shaded region will work. For example, 6 children and 1 adult or 1 child and 5 adults.
Explanation:
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