Question

1 a population of ants was 10,000 at the start of april. since then, it has tripled each month.
a. complete the table.

months since april	number of ants
1
2
3
4

b. what do you notice about the population differences from month to month?
c. if there are n ants one month, how many ants will there be a month later?
2 a swimming pool contains 500 gallons of water. a hose is turned on, and it fills the pool at a rate of 24 gallons per minute. which expression represents the amount of water in the pool, in gallons, after 8 minutes?
a 500·24·8
b 500+24+8
c 500+24·8
d 500·24⁸
3 the population of a city is 100,000. it doubles each decade for 5 decades. select all expressions that represent the population of the city after 5 decades.
a 32,000
b 320,000
c 100,000·2·2·2·2·2
d 100,000·5²
e 100,000·2⁵

Explanation:

Step1: Calculate initial ant population

At 0 months, population = $10,000$

Step2: Calculate month 1 ant population

Triple the initial value: $10,000 \times 3 = 30,000$

Step3: Calculate month 2 ant population

Triple month 1 value: $30,000 \times 3 = 90,000$

Step4: Calculate month 3 ant population

Triple month 2 value: $90,000 \times 3 = 270,000$

Step5: Calculate month 4 ant population

Triple month 3 value: $270,000 \times 3 = 810,000$

Step6: Analyze monthly population differences

Find differences: $30,000-10,000=20,000$; $90,000-30,000=60,000$; $270,000-90,000=180,000$; $810,000-270,000=540,000$. Observe the pattern of these differences.

Step7: Define population for next month

If current population is $n$, triple it for next month: $n \times 3 = 3n$

Step8: Identify pool water expression

Initial water + (rate × time): $500 + 24 \times 8$

Step9: Calculate city population after 5 decades

Initial population × $2^5$: $100,000 \times 2 \times 2 \times 2 \times 2 \times 2 = 3,200,000$; match to valid expressions.

Answer:

1. a.

Months Since April	Number of Ants
1	30,000
2	90,000
3	270,000
4	810,000

1. b. The population differences triple each month.
2. c. $3n$
3. C. $500 + 24 \cdot 8$
4. B. 3,200,000; C. $100,000 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$; E. $100,000 \cdot 2^5$