Questions: Suppose the total value for a cryptocurrency (in billions of dollars) was 103 in January 2020 and 755 in January 2021. Complete parts (a) through (c). (a) Find an exponential function of the form f(t)=y0 b^t to model these data with t in months, where t=0 corresponds to January 2020. f(t)=103 * 1.181^t (Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.) (b) If the model remains accurate, what would have been the predicted value for May 2021? If the model remained accurate, the predicted value would have been 1467 billion. (Round to the nearest whole number as needed.) (c) By experimenting with different values (or using a graphing calculator to solve an appropriate equation), estimate the first full month in which the value of the cryptocurrency exceeded 460 billion. The first full month in which the value exceeded 460 was in of the year (Type a whole number.)

Solution Steps

To solve these problems, we need to model the growth of the cryptocurrency using an exponential function. For part (a), we determine the base of the exponential function using the given values for January 2020 and January 2021. For part (b), we use the exponential function to predict the value in May 2021. For part (c), we find the first month where the value exceeds $460 billion by solving the inequality using the exponential model.

To model the growth of the cryptocurrency, we use the exponential function of the form:

\[ f(t) = y_0 b^t \]

where \( y_0 = 103 \) (the value in January 2020) and \( b \) is calculated as follows:

\[ b = \left( \frac{y_1}{y_0} \right)^{\frac{1}{t_1}} = \left( \frac{755}{103} \right)^{\frac{1}{12}} \approx 1.181 \]

Thus, the exponential function is:

\[ f(t) = 103 \cdot 1.181^t \]

To find the predicted value for May 2021, we calculate \( f(t) \) for \( t = 16 \) (16 months from January 2020):

\[ f(16) = 103 \cdot 1.181^{16} \approx 1466.6175 \]

Rounding to the nearest whole number gives:

\[ \text{Predicted value for May 2021} \approx 1467 \text{ billion} \]

To find the first full month in which the value exceeds $460 billion, we solve the inequality:

\[ f(t) > 460 \]

This leads to:

\[ 103 \cdot 1.181^t > 460 \]

Taking logarithms, we find:

\[ t > \frac{\log\left(\frac{460}{103}\right)}{\log(1.181)} \approx 10 \]

Thus, the first full month where the value exceeds $460 billion is:

\[ \text{Month} = 10 \]

Final Answer