Calculate the future cost of a commodity, the annual rate of compound interest, and the amount in an account after different times.

1. Calculate the future cost of the commodity.

Step 1: Identify the given values.
Initial cost $P = \text{M}500$
Inflation rate $r = 10\% = 0.10$
Time $t = 5$ years

Step 2: Use the future value formula for inflation.

$$FV = P(1 + r)^t$$

Step 3: Substitute the values and calculate.

$$FV = \text{M}500(1 + 0.10)^5$$ $$FV = \text{M}500(1.10)^5$$ $$FV = \text{M}500(1.61051)$$ $$FV = \text{M}805.255$$

The commodity will cost M805.26 in 5 years.

$$\boxed{\text{M}805.26}$$

2. Calculate the annual rate of compound interest.

Step 1: Identify the given values.
Initial amount $P = \text{M}2,000$
Future amount $FV = \text{M}2,721$
Time $t = 4$ years

Step 2: Use the compound interest formula and solve for $r$.

$$FV = P(1 + r)^t$$ $$\text{M}2,721 = \text{M}2,000(1 + r)^4$$

Step 3: Isolate $(1 + r)^4$.

$$\frac{2,721}{2,000} = (1 + r)^4$$ $$1.3605 = (1 + r)^4$$

Step 4: Take the fourth root of both sides.

$$(1 + r) = (1.3605)^{\frac{1}{4}}$$ $$(1 + r) \approx 1.08$$

Step 5: Solve for $r$.

$$r = 1.08 - 1$$ $$r = 0.08$$ $$r = 8\%$$

The annual rate of compound interest is 8%.

$$\boxed{8\%}$$

3. Calculate the amount in Peo's account at different times.

Initial deposit $P = \text{M}3,500$
Interest rates:
$r_1 = 3.7\% = 0.037$ for the first 4 years
$r_2 = 4.2\% = 0.042$ for the next 4 years (years 5-8)
$r_3 = 4.8\% = 0.048$ for the final 2 years (years 9-10)