1.) there are 8 people in a room.
each person shakes hands with each of the other people once and only once.
how many handshakes take place?

2.) a student has some pigs and some chickens.
he finds that together they have 70 heads and 200 legs.
how many pigs and how many chickens does he have?

3.) a well is 10 feet deep.
a frog climbs up 5 feet during the day, but falls back 4 feet at night.
assuming that the frog starts at the bottom of the well,
how many days does it take the frog to get to the top?

i'm looking for work shown, too, ! : )

1. 8 people, every person would shake hands with 7 other people
1st person shakes hands with 7 people
2nd person shakes hands with 6 people ( they already shook hands with the first person)
3rd person shakes 5 hands
4th shakes 4 hands
5th person shakes 3 hands
6th person shakes 2 hands
7th person shakes 1 other hand
 total = 7 +6 +5 +4 +3 +2 1 = 28 handshakes total

2. P = 70-C
4P + 2C = 200
4(70-C) +2C = 200
280 - 4c + 2c = 200
280 -2c = 200
-2c = -80
c = -80 / -2 = 40 chickens 
70-40 = 30 pigs

3. 6 days = see picture:

I’m confused what’s happening

a. geometricb. g[h] = (3/4)·g[h-1]; g[1] = 384c. g[h] = 384·(3/4)^(h-1)d. g[12] ≈ 16.2183 gallonse. g[12] ≈ 64.8732 quarts

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt