Math / Calculus

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DETAILS MY NOTES TANAPMATH7 3.3.005. PRACTICE ANOTHER 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 $2 \% / y e a r$ , find the length of time $t_{4}$ (in yr) required for the population to quadruple in size. $t_{4}=$ $\square$ yr (b) Using the time $t_{4}$ found in part (a), what would be the world population (in billions of people) if the growth rate were reduced to $1.7 \% / y r$ ? $\square$ billion people Need Help? Read it Watch it

Studdy Solution

STEP 1

1. The population grows exponentially.
2. The initial population size is $P_0$ .
3. The growth rate is $2\%$ per year for part (a) and $1.7\%$ per year for part (b).

STEP 2

1. Determine the formula for exponential growth.
2. Calculate the time required for the population to quadruple.
3. Calculate the population size with a reduced growth rate.

STEP 3

Recall the formula for exponential growth:
$P(t) = P_0 \cdot e^{rt}$
where $P(t)$ is the population at time $t$ , $P_0$ is the initial population, $r$ is the growth rate, and $t$ is the time in years.

STEP 4

To find the time $t_4$ required for the population to quadruple, set $P(t_4) = 4P_0$ and solve for $t_4$ :
$4P_0 = P_0 \cdot e^{0.02t_4}$
Divide both sides by $P_0$ :
$4 = e^{0.02t_4}$
Take the natural logarithm of both sides:
$\ln(4) = 0.02t_4$
Solve for $t_4$ :
$t_4 = \frac{\ln(4)}{0.02}$
Calculate $t_4$ and round to two decimal places:
$t_4 \approx \frac{1.3863}{0.02} \approx 69.32 \text{ years}$

STEP 5

Using the time $t_4 \approx 69.32$ years, calculate the population if the growth rate is reduced to $1.7\%$ :
$P(t_4) = P_0 \cdot e^{0.017 \times 69.32}$
Calculate the exponent:
$0.017 \times 69.32 \approx 1.17844$
Calculate $P(t_4)$ :
$P(t_4) = P_0 \cdot e^{1.17844}$
$P(t_4) \approx P_0 \cdot 3.25$
The population would be approximately $3.25$ times the initial population, in billions.
The time required for the population to quadruple is:
$\boxed{69.32} \text{ years}$
The population with a reduced growth rate would be approximately:
$\boxed{3.25} \text{ billion people}$

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