Question

The population of a community is known to increase at a rate proportional to the number of people present at time t. if an initial population p0 has doubled in 7 years, how long will it take to triple? (round your answer to one decimal place.)

Answer 1

To determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(rt) for t. Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.

Step-by-step explanation:

To solve this problem, we can use the equation P = P0e^(rt), where P is the population at a given time, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), r is the growth rate, and t is the time in years.

In this case, the population has doubled in 7 years, so we can write the equation as 2P0 = P0e^(r*7). We need to solve for r.

Dividing both sides of the equation by P0 gives us 2 = e^(7r). Taking the natural logarithm of both sides gives us ln(2) = 7r. Solving for r, we find that r ≈ ln(2)/7.

Now, to determine how long it will take for the population to triple, we need to solve the equation 3P0 = P0e^(r*t) for t. Substituting the value of r we found, we get 3 = e^((ln(2)/7)*t). Taking the natural logarithm of both sides to solve for t, we have ln(3) = (ln(2)/7)*t. Solving for t, we find that t ≈ 7 * ln(3)/ln(2). Rounding the answer to one decimal place, it will take approximately 10.1 years for the population to triple.

Answer 2

11.1 years The phrase "increase at a rate proportional to the number of people present" is a rather wordy way is saying "the population grows at a geometric rate". Or "the growth rate is exponential" and that last statement is the critical one. Let's take the log of 2 and divide by 7 log(2)/7 = 0.30103 / 7 = 0.043004 Now if you calculate 10^0.043004, you'll get 1.10409 which is how much the population is increasing each year. But you don't need that number. Instead, take the log of 3 and divide by the 0.043004 figure you got earlier. log(3)/0.043004 = 0.477121255/0.043004 = 11.09474 So it will take 11.1 years for the population to triple. Let's check that by raising 1.10409 to the 7th and 11.1 powers. 1.10409^7 = 2.0000 The above confirms that the rate of 1.10409 properly doubles the population in 7 years. 1.10409^11.1 = 3.00 And that same growth rate will triple the population in 11.1 years.