These should be easy but I'm having a problem figuring where to start, any advice?

Question

A) The population of China is growing exponentially at a rate of 1.1% per year.
How long will it take China to double its population? _________________

B) The value of the property in S-ville decreases exponentially at a rate of 6.8% annually. The original value of the house was $150,000.
Write the function modeling the value of the house over time.
V(t) = _________________________________

Answer 1

A) In this question, it would be helpful to declare some variables. For example, let P be the population of China initially. Therefore, double the population would mean 2P.

This question is a geometric progression, meaning that each term of the progression follows a fixed ratio. In this question. the ratio, denoted as r, will be 1.011. Notice that the ratio is greater than 1; this means that the terms are increasing in value, which is correct since the population of China is growing,.

For a geometric progression, the nth term, denoted as u with subscript n) can be calculated using the formula

u(subscript n) = a x r^(n-1) where a is the first term in the progression, r is the ratio and n is the number at which the term you want to find is in, In this question, they are asking us to find n, given u(subscript n), a and r. In this case:

The term we want is double the population or, 2P. The first term is P, while the common ratio is 1.011. Substitute this into the formula and we get:

2P = P x 1.011^(n-1)
1.011^(n-1) = 2

Here, we take logarithms base 10 on both sides.

lg [1.011^(n-1)] = lg 2
(n-1) lg 1.011 = lg 2
n-1 = 63.53593
n = 64.53593

Rounding off to 3 significant figures, the population of China will double in 64.5 years. (Ans)

B) Here, I let V = 150000. In this question the value of each consecutive term is decreasing, so the common ratio must be less than 1.

Now, the value of the house decreases by 6.8% every year. This means that at the end of a particular year, the value of the house is 93.2% of what it was at the beginning of that year. This will continue for the many years to come; the value of the house in a year is 93.2% or 0.932 of the value in the preceding year. So, the common ratio is 0.932.

V(t) here is defined as the value of the house at a year t. For example, if you want to know the value of the house in the second year, then substitute 2 for t, then find what is V(2).

Using the formula to find the nth term of a geometric progression, we get:

V(t) = 150000 x 0.932^t (Ans)

Notice that there is no "t-1". This is because I define the first year as the starting year, where the value of the house was still $150000.

Answer 2

(1+0.011) ^n = 2 so n= log2/ log(1.011)=63.36 years
V = (1-0.068)^t *150,000