11 and 13 only11. the sum s of the first n consecutive positive integers, 1, 2, 3, , n, is given by the formula s=0.5n(n+1). find n when s=120 and check your answer by addition.13. a boat travels downstream (with the current) for 36 miles and then makes the return trip upstream (against the current). the trip downstream took 3/4 of an hour less than the upstream trip. if the rate of the current is 4 mph, find the rate of the boat in still water and the time for each part of the trip.

11. The sum S of the first n consecutive positive integers, 1, 2, 3, ... , n, is given by the formula S=0.5n(n+1). Find n when S=120 and check your answer by addition.

We need two numbers whose sum is 1 and multiply to -240, so two numbers whose difference is 1 and multiply to 240, 15 and 16. We factor:

We reject the negative root.

n=15

Check:

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15

=1+15 + 2+14 + 3+13 + 4+12 + 5+11 +6+10 + 7+9 + 8

= 7(16)+8

= 120 good

13. A boat travels downstream (with the current) for 36 miles and then makes the return trip upstream (against the current). The trip downstream took 3/4 of an hour less than the upstream trip. If the rate of the current is 4 mph, find the rate of the boat in still water and the time for each part of the trip.

Let v be the speed of the boat in still water.

The speed downstream is v+4 so the time to travel 36 miles downstream is 36/(v+4)

The speed upstream is v-4 so the time to travel 36 miles upstream is 36/(v-4)

Writing our equation,

3/4 = 36/(v-4) - 36/(v+4)

3 = 144 (1/(v-4) - 1/(v+4))

3(v-4)(v+4) = 144( (v+4) - (v-4) )

v^2 - 16 = 48(8)

v^2 = 48(8) + 16 = 400

v = 20 mph, the rate of the boat in still water

Time downstream is 36/(20+4)=36/24 = 3/2 = 1.5 hours

Time upstream is 36/(20-4)=36/16=9/4 = 2.25 hours

3/4 hour difference, good

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: