Chapter 1 Text work review

9. Fan out system:

t₁ = 1, r = 4

a) 1 + 4 + 16 + 64 + ...   (if person in charge is included)

    4 + 16 + 64 + ...   (if person in charge is not included)

b) Let's go with the person in charge not being included, and find out S₁₀.

In this case, t₁ = 4, r = 4:

S₁₀ = 4(4¹⁰ - 1)/(4 - 1) = 1,398,100 people contacted.

10. 

The ball will drop 6 times, that is, n = 6, where t₁ = 20, and r = 0.4.

The ball will rise 5 times (n = 5), with t₁ = 0.4(20) = 8, and r = 0.4.

To find the total vertical distance travelled, there are a few different ways we could look at this.

First, we could find the sum of the two series: down and up:

S_down6 + S_up5 = 20(0.4⁶ - 1)/(0.4 - 1) + 8(0.4⁵ - 1)/(0.4 - 1) = 33.2 + 13.2 = 46.4m

Or... since after the first drop, the distance up is the same as the distance down, we could look at ...

20 + 2S₅ where t₁ = 8, r = 0.4 --> distance = 20 + 2\( \cdot \)8(0.4⁵ - 1)/(0.4 - 1) = 20 + 2\( \cdot \)13.2 = 46.4m

11. Marathon training: t₁ = 25km, r = 1.1, n = 15, S₁₅ = ?

S₁₅ = 25(1.1¹⁵ - 1)/(1.1 - 1) = 794.3km

12. Koch snowflakes:

length: t_n = (\( \frac{1}{3} \))^{n - 1}

# segments: t_n = 3(4)^{n - 1}

perimeter: t_n = 3(\( \frac{4}{3} \))^{n - 1}

d) perimeter at stage 6 (n = 6)

t₆ = 3(\( \frac{4}{3} \))^{6 - 1} = \( \frac{1024}{81} \) = 12.64

13. Advertising company: t₁ = 1000, r = 1.4, want S₁₁.

n is a counter and when n = 2, it represents the next 10 days. We want 100 days, so n = 11:

S₁₁ = 1000(1.4¹¹ - 1)/(1.4 - 1) = 98739 people.

14. 10 beads: t = 24mm, r = 0.75, n = 10, S₁₀ = ?

S₁₀ = 24(0.75¹⁰ - 1)/(0.75 - 1) = 91mm

15. 200mg ampicillan ear drugs: t₁ = 200, r = 0.12

a) S₃ = 200(0.12³ - 1)(/(0.12 - 1) = 226.9mg

b) S₆ = 200(0.12⁶ - 1)/(0.12 - 1) = 227.3mg