Math Adventures/Double check/Answers

1. 2 holes.
8 men → 8 holes in 8 days ⇒ team rate = 1 hole/day ⇒ per-man rate = 1/8 hole/day.
4 men × 4 days × (1/8) = 2.

2. $0.05.
Let ball = x, bat = x + 1.00; x + x + 1.00 = 1.10 ⇒ x = $0.05.

3. $20 profit.
Net = −60 + 70 − 80 + 90 = +20.

4. 2 km/h current.
Downstream 30/3 = 10 km/h; upstream 30/5 = 6 km/h.
v + c = 10, v − c = 6 ⇒ c = 2 km/h.

5. 1/3 (given “at least one is a boy”)
Outcomes {BB, BG, GB} are equally likely ⇒ BB is 1/3.
(If instead “the older is a boy,” it would be 1/2.)

6. 2 flips.
Geometric with p = 1/2 ⇒ E[N] = 1/p = 2.

7. Cannot be determined from the information given.
A 10-ft ladder with the bottom sliding 3 ft needs the initial foot distance to compute the drop:
height changes from √(100 − x²) to √(100 − (x + 3)²).

8. 8 pieces.
Max pieces from n planar cuts: (n³ + 5n + 6)/6 ⇒ for n = 3 gives (27 + 15 + 6)/6 = 8.

9. It depends on the individuals’ crossing times.
For four people with times t₁ ≤ t₂ ≤ t₃ ≤ t₄, the optimal total is
min( t₁ + 2t₂ + t₄, 2t₁ + t₃ + t₄ ).
(Example: with 1, 2, 7, 10 minutes, minimum = min(1 + 4 + 10, 2 + 7 + 10) = 17.)