Question

dead how long until the zombies take over? on the show, the walking dead, a disease was contracted that turns people into zombies called walkers. if a walker bites a human, the human is turned into a walker. assume that each walker turns one person a week into a zombie and that none of the walkers are killed. 1. the diagram below represents a town with 140 people. each box represents a human; each crossed out box represents a walker. keep track of the human and walker populations over time. grid image chart with weeks (0 - 10) and columns for walkers (1 at week 0) and humans (139 at week 0) a. how does the walker population change each week? how long will it take before everyone in town is a walker? b. list the number of walkers as a sequence for the first 7 weeks. what type of sequence is this? c. write a recursive formula for the sequence. d. graph the relationship between the week and the number of walkers. graph grid image e. is the relationship linear? how do you know?

Explanation:

Step1: Analyze Walker population growth

Each Walker turns 1 person into a Walker per week, so the Walker population doubles each week? Wait, no—wait, initial Walker is 1. In week 1, 1 Walker turns 1 human, so Walkers become \(1 + 1=2\)? Wait, no, wait: each Walker turns one person. So if there are \(W\) Walkers, next week Walkers are \(W + W=2W\)? Wait, no, initial: week 0, Walkers = 1, Humans = 139. Week 1: each Walker (1) turns 1 human, so Walkers become \(1 + 1 = 2\), Humans become \(139 - 1 = 138\). Week 2: Walkers = \(2 + 2 = 4\), Humans = \(138 - 2 = 136\). Wait, no, that's doubling. Wait, but let's check the total population: Walkers + Humans should be 140. So week 0: 1 + 139 = 140. Week 1: 2 + 138 = 140. Week 2: 4 + 136 = 140. Week 3: 8 + 132 = 140? Wait, no, 4 Walkers would turn 4 humans, so Walkers become 4 + 4 = 8, Humans 136 - 4 = 132. Yes. So the Walker population is a geometric sequence with first term \(a_1 = 1\), common ratio \(r = 2\).

To find when everyone is a Walker, we need \(W = 140\). So solve \(2^{n - 1}=140\)? Wait, week 0: \(2^0 = 1\), week 1: \(2^1 = 2\), week 2: \(2^2 = 4\),..., week \(n\): \(2^n\) Walkers? Wait, week 0: \(n=0\), \(2^0 = 1\); week 1: \(2^1 = 2\); so general term \(W(n)=2^n\) for \(n\) weeks (starting at 0). We need \(2^n = 140\). Let's compute \(2^7 = 128\), \(2^8 = 256\). Wait, that can't be. Wait, no, my mistake: initial Walker is 1 (week 0). Week 1: 2, week 2: 4, week 3: 8, week 4: 16, week 5: 32, week 6: 64, week 7: 128, week 8: 256. But total population is 140, so when Walkers reach 140? Wait, no—Wait, maybe I messed up the growth. Wait, each Walker turns one person, so the number of new Walkers is equal to the current number of Walkers. So the recurrence is \(W_{n+1}=W_n + W_n = 2W_n\), with \(W_0 = 1\). So \(W_n = 2^n\). We need \(W_n = 140\). But \(2^7 = 128\), \(2^8 = 256\), which is more than 140. But total population is 140, so when Walkers + Humans = 140, so when Humans = 0, Walkers = 140. So solve \(140 - W_n = 0\)? Wait, no: Humans start at 139, and each week, Humans decrease by \(W_n\) (since each Walker turns one human). Wait, no: in week 0, Humans = 139, Walkers = 1. Week 1: Humans = 139 - 1 = 138, Walkers = 1 + 1 = 2. Week 2: Humans = 138 - 2 = 136, Walkers = 2 + 2 = 4. Week 3: Humans = 136 - 4 = 132, Walkers = 4 + 4 = 8. Week 4: Humans = 132 - 8 = 124, Walkers = 8 + 8 = 16. Week 5: Humans = 124 - 16 = 108, Walkers = 16 + 16 = 32. Week 6: Humans = 108 - 32 = 76, Walkers = 32 + 32 = 64. Week 7: Humans = 76 - 64 = 12, Walkers = 64 + 64 = 128. Week 8: Humans = 12 - 128 = negative? Wait, that can't be. Wait, no—when Walkers are 128, Humans are 12. Then in week 8, Walkers would turn 128 humans, but there are only 12 humans left. So actually, the growth is limited by the total population. Wait, the problem says "none of the Walkers are killed", so when there are \(W\) Walkers, they can turn \(W\) humans, but if \(W >\) Humans, then all remaining humans are turned, and Walkers become \(W +\) Humans. Wait, I made a mistake earlier. Let's correct:

Total population \(T = 140\) (Walkers + Humans). Let \(W_n\) be Walkers in week \(n\), \(H_n\) be Humans. Then \(W_0 = 1\), \(H_0 = 139\).

For \(n \geq 0\), \(W_{n+1}=W_n + \min(W_n, H_n)\), and \(H_{n+1}=H_n - \min(W_n, H_n)\). But since initially \(W_n < H_n\) (week 0: 1 < 139), so \(\min(W_n, H_n)=W_n\), so \(W_{n+1}=2W_n\), \(H_{n+1}=H_n - W_n\).

But when \(W_n \geq H_n\), then \(W_{n+1}=W_n + H_n = T\), \(H_{n+1}=0\).

So let's compute step by step:

Week 0: \(W=1\), \(H=139\)

Week 1: \(W=1+1=2\), \(H=139-1=138\)

Week 2: \(W=2+2=4\), \(H=138-2=136\)

Week 3: \(W=4+4=8\), \(H=136-4=132\)

Week 4: \(W=8+8=16\), \(H=132-8=124\)

Week 5: \(W=16+16=32\), \(H=124-16=108\)

Week 6: \(W=32+32=64\), \(H=108-32=76\)

Week 7: \(W=64+64=128\), \(H=76-64=12\)

Week 8: Now, \(W=128\), \(H=12\). Since \(W > H\), so \(W_{8}=128 + 12=140\), \(H_{8}=0\). Ah, so in week 8, all are Walkers.

Wait, let's check the total: week 7: \(W=128\), \(H=12\) (128+12=140). Week 8: \(W=128+12=140\), \(H=0\). So it takes 8 weeks? Wait, but let's check the doubling: week 0:1, week1:2, week2:4, week3:8, week4:16, week5:32, week6:64, week7:128, week8:140. Wait, week7:128 Walkers, 12 Humans. Then week8: 128 Walkers turn 12 Humans, so Walkers become 128+12=140, Humans 0. So the Walker population changes by doubling each week until week7, then in week8, it increases by the remaining Humans.

But the question is "how does the Walker population change each week?" Initially, it doubles (since \(W_{n+1}=2W_n\) when \(W_n < H_n\)). Then, when \(W_n \geq H_n\), it increases by \(H_n\) to reach 140.

But to find when everyone is a Walker: we need \(H_n = 0\). Let's see the Human population: \(H_n = 140 - W_n\). So we need \(140 - W_n = 0 \implies W_n = 140\).

From the sequence, \(W_0=1\), \(W_1=2\), \(W_2=4\), \(W_3=8\), \(W_4=16\), \(W_5=32\), \(W_6=64\), \(W_7=128\), \(W_8=140\) (since \(W_7=128\), \(H_7=12\), so \(W_8=128+12=140\)).

So the Walker population doubles each week (geometric sequence with ratio 2) until week 7, then in week 8, it reaches 140. So the time to take over is 8 weeks? Wait, but let's check the total population. Wait, maybe my initial assumption of doubling was wrong because when Walkers exceed Humans, the growth changes. But let's check the table:

Wait, the table has weeks 0 to 10. Let's fill the table:

Week 0: Walkers=1, Humans=139

Week 1: Walkers=2, Humans=138

Week 2: Walkers=4, Humans=136

Week 3: Walkers=8, Humans=132

Week 4: Walkers=16, Humans=124

Week 5: Walkers=32, Humans=108

Week 6: Walkers=64, Humans=76

Week 7: Walkers=128, Humans=12

Week 8: Walkers=140, Humans=0

So in week 8, Humans=0, so everyone is a Walker.

So for part a: The Walker population doubles each week (geometric growth) until it reaches 140, which takes 8 weeks.

Step2: For part b: First 7 weeks (weeks 0 to 6? Wait, "first 7 weeks"—week 0 is start, so weeks 0-6? Wait, the question says "for the first 7 weeks"—so week 0 (1), week 1 (2), week 2 (4), week 3 (8), week 4 (16), week 5 (32), week 6 (64). Wait, week 7 would be 128, but first 7 weeks: weeks 0 to 6 (7 terms: 0-6). So the sequence is 1, 2, 4, 8, 16, 32, 64. This is a geometric sequence with first term 1 and common ratio 2.

Step3: For part c: Recursive formula. Since \(W_{n+1}=2W_n\) for \(n\) where \(W_n < H_n\), and \(W_0=1\). So recursive formula: \(W_0 = 1\), \(W_{n+1}=2W_n\) for \(n \geq 0\) (until \(W_n \geq H_n\), but since we can write the general recursive for the geometric part: \(W_{n} = 2 \cdot W_{n - 1}\) with \(W_0 = 1\).

Step4: For part d: To graph, we have weeks on x-axis (0-10) and Walkers on y-axis. The points are (0,1), (1,2), (2,4), (3,8), (4,16), (5,32), (6,64), (7,128), (8,140), (9,140), (10,140). So it's an exponential curve until week 8, then flat at 140.

Step5: For part e: The relationship is not linear. A linear relationship has a constant slope (constant rate of change). Here, the rate of change of Walkers is \(W_{n+1}-W_n = W_n\) (since \(W_{n+1}=2W_n\), so change is \(W_n\)), which increases each week (1, 2, 4, 8,...), so the slope is not constant. Hence, it's exponential, not linear.

But let's focus on part a first:

Part a:

How does Walker population change each week? Initially, it doubles each week (since each Walker turns one human, so Walkers become \(W + W = 2W\)). How long until everyone is Walker? We need \(W = 140\). From the sequence:

Week 0: 1

Week 1: 2

Week 2: 4

Week 3: 8

Week 4: 16

Week 5: 32

Week 6: 64

Week 7: 128

Week 8: 128 + 12 = 140 (since Humans at week 7 are 12, so Walkers turn all 12, making Walkers 140, Humans 0). So it takes 8 weeks.

Wait, but let's check the total population: 1 (week0) + 139 = 140. Week1: 2 + 138 = 140. Week2: 4 + 136 = 140. Week3: 8 + 132 = 140. Week4: 16 + 124 = 140. Week5: 32 + 108 = 140. Week6: 64 + 76 = 140. Week7: 128 + 12 = 140. Week8: 140 + 0 = 140. So at week8, Humans=0, so everyone is a Walker.

So the Walker population doubles each week (geometric growth) until week7, then in week8, it reaches 140. The time to take over is 8 weeks.

Answer (for part a):

The Walker population doubles each week (geometric growth, \(W_{n+1} = 2W_n\) initially). It takes 8 weeks for everyone in town to be a Walker.

Answer:

(for part a):
The Walker population doubles each week (geometric growth, \(W_{n+1} = 2W_n\) initially). It takes 8 weeks for everyone in town to be a Walker.