The world population at the beginning of 1990 was 5.3 billion. Assume that the population continues to grow at the rate of approximately 2%/year and find the function Q(t) that expresses the world population (in billions) as a function of time t (in years), with t = 0 corresponding to the beginning of 1990. (Round your answers to two decimal places.)

(a). If the world population continues to grow at approximately 2%/year, find the length of time t0 required for the population to double in size.

t0=____yr

(b). Using the time t0 found in part (a), what would be the world population if the growth rate were reduced to 1.2%/yr?

____ billion people

I don't understand how to set this problem up correctly.

Can someone please explain?

a. The formula for exponential growth is P(t) = P0*e^(rt), where P0 is the initial population, r is the growth rate, and t is the time. In this case, P0 = 5.3 and r = 0.02. To find t0, we need to solve for t when P(t) = 2P0. This gives us t0 = ln(2)/(0.02) = 34.53 yr.

b. To find the population with a growth rate of 1.2%, we can use the same formula, but with r = 0.012. This gives us P(t) = 5.3*e^(0.012*t). To find the population after 34.53 years, we plug in t = 34.53 and get P(34.53) = 5.3*e^(0.412) = 10.6 billion people.

To find the function Q(t) that expresses the world population as a function of time t, we can use the formula for exponential growth:

Q(t) = Q0 * (1 + r)^t

Where Q(t) is the population at time t, Q0 is the initial population, r is the growth rate (expressed as a decimal), and t is the time in years.

(a). To find the length of time t0 required for the population to double in size, we need to solve the equation Q(t0) = 2 * Q0.

Let's plug in the given values:
Q(t0) = 2 * 5.3 billion
Q(t0) = 10.6 billion

Now, we substitute Q(t) from the formula for exponential growth:

Q(t0) = Q0 * (1 + r)^t0
10.6 billion = 5.3 billion * (1 + 0.02)^t0

Now we need to solve for t0. Taking the logarithm of both sides can help us isolate the exponent:

ln(10.6 billion/5.3 billion) = ln(1.02^t0)

Simplifying further:
ln(2) = ln(1.02^t0)
ln(2) = t0 * ln(1.02)

Dividing both sides by ln(1.02):
t0 = ln(2) / ln(1.02)

Using a calculator, we can find the value of t0:
t0 ≈ 35.00 years

Therefore, it would take approximately 35 years for the world population to double.

(b). Using the time t0 found in part (a), we can calculate the world population if the growth rate were reduced to 1.2% per year.

Using the formula for exponential growth:

Q(t) = Q0 * (1 + r)^t

Given that Q0 = 5.3 billion, r = 0.012 (1.2% expressed as a decimal), and t = t0 = 35 years, we can substitute these values into the equation:

Q(t) = 5.3 billion * (1 + 0.012)^35

Calculating this value:
Q(t) ≈ 5.3 billion * (1.012)^35 ≈ 5.3 billion * 1.62

Q(t) ≈ 8.58 billion

Therefore, if the growth rate were reduced to 1.2%/yr, the world population would be approximately 8.58 billion people.

To solve this problem, you need to set up an exponential growth function to represent the world population. The general form of an exponential growth function is:

Q(t) = Q₀ * (1 + r)^t

Where:
- Q(t) represents the world population at time t.
- Q₀ represents the initial world population at t = 0.
- r represents the growth rate (as a decimal) per unit of time.
- t represents the time (in years).

Given that the world population at the beginning of 1990 was 5.3 billion, we can substitute Q₀ = 5.3 into the formula. The growth rate is approximately 2% per year, or 0.02 as a decimal:

Q(t) = 5.3 * (1 + 0.02)^t

Now, let's solve part (a) of the problem, which asks for the time required for the population to double.

To double the population, the final population (Q(t)) would be twice the initial population (Q₀). Therefore:

2 * Q₀ = Q₀ * (1 + 0.02)^t₀

Dividing both sides of the equation by Q₀:

2 = (1 + 0.02)^t₀

To solve for t₀, we need to take the logarithm of both sides of the equation. Using natural logarithms (ln):

ln(2) = ln((1 + 0.02)^t₀)

Applying the logarithmic property ln(a^b) = b * ln(a):

ln(2) = t₀ * ln(1 + 0.02)

Now, isolate t₀:

t₀ = ln(2) / ln(1.02)

Using a calculator, we can plug in the values and calculate t₀. Rounded to two decimal places, t₀ ≈ 34.66 years.

Now moving on to part (b) of the problem. Given the value of t₀, we need to find the world population if the growth rate is reduced to 1.2% per year.

We can use the same formula:

Q(t) = 5.3 * (1 + 0.02)^t

But now we substitute r = 0.012 (1.2% as a decimal) and t = t₀:

Q(t₀) = 5.3 * (1 + 0.012)^34.66

Calculate this expression using a calculator to find the world population when the growth rate is reduced to 1.2% per year.