Question

a. identify the continuous rate and use put it in a sentence to qualify the change of the output b. compute the associated rate per unit input (3 dec places for rate) and write down the equation 2. u.s. population

Explanation:

Part (a)

Step1: Recall the continuous growth formula

The general form of a continuous growth model is \( y = ae^{rx} \), where \( a \) is the initial amount, \( r \) is the continuous growth rate, and \( x \) is the independent variable (usually time).

Step2: Identify the continuous rate from the given equation

The given equation for the U.S. population is \( y = 0.0036e^{0.0126x} \). Comparing this with the general form \( y = ae^{rx} \), we can see that the continuous growth rate \( r \) is \( 0.0126 \) or \( 1.26\% \).

Step3: Qualify the change in a sentence

The U.S. population in the 20th century follows a continuous growth model \( y = 0.0036e^{0.0126x} \), where the population grows continuously at a rate of \( 1.26\% \) per unit of time (where \( x \) represents the time variable, likely years since a certain starting point, e.g., 1900).

Part (b)

Step1: Recall the relationship between continuous rate and annual growth factor

For a continuous growth model \( y = ae^{rx} \), the associated annual growth factor (when \( x \) is in years) is given by \( e^{r} \), where \( r \) is the continuous growth rate.

Step2: Calculate the annual growth factor

We know that \( r = 0.0126 \). So we need to compute \( e^{0.0126} \). Using a calculator, \( e^{0.0126} \approx 1.01268 \) (rounded to 5 decimal places). To three decimal places, we calculate \( e^{0.0126} \):

First, recall that \( e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} \) for small \( x \). Let \( x = 0.0126 \).

\( x = 0.0126 \)

\( x^2 = (0.0126)^2 = 0.00015876 \)

\( x^3 = (0.0126)^3 = 0.000001999376 \)

\( e^{0.0126} \approx 1 + 0.0126 + \frac{0.00015876}{2} + \frac{0.000001999376}{6} \)

\( = 1 + 0.0126 + 0.00007938 + 0.000000333 \)

\( \approx 1.012679713 \approx 1.013 \) (to three decimal places)

Step3: Write the equation for the factor

The annual growth factor \( b \) is related to the continuous rate \( r \) by \( b = e^r \). So with \( r = 0.0126 \), the equation for the factor is \( b = e^{0.0126} \approx 1.013 \) (to three decimal places).

Answer:

s:

Part (a)

The continuous growth rate is \( 0.0126 \) (or \( 1.26\% \)). The U.S. population in the 20th century grows continuously at a rate of \( 1.26\% \) per unit time (years) according to the model \( y = 0.0036e^{0.0126x} \).

Part (b)

The annual growth factor (to three decimal places) is approximately \( 1.013 \). The equation for the factor is \( b = e^{0.0126} \), and when calculated, \( b \approx 1.013 \).