Questions: (a) Let t be the time (in days) since the beginning of the study, and let y be the number of bacteria at time t. Write a formula relating y to t. Use exact expressions to fill in the missing parts of the formula. Do not use approximations. y=20 e^(ln 13/52 t) (b) How many bacteria are there 17 days after the beginning of the study? Do not round any intermediate computations, and round your answer to the nearest whole number. bacteria

Solution Steps

(a) The formula relating the number of bacteria \( y \) to the time \( t \) is already given as: \[ y = 20 e^{\left(\frac{\ln 13}{52}\right) t} \]

(b) To find the number of bacteria 17 days after the beginning of the study, we need to substitute \( t = 17 \) into the given formula and compute the value of \( y \).

The formula relating the number of bacteria \( y \) to the time \( t \) is: \[ y = 20 e^{\left(\frac{\ln 13}{52}\right) t} \]

To find the number of bacteria 17 days after the beginning of the study, substitute \( t = 17 \) into the formula: \[ y = 20 e^{\left(\frac{\ln 13}{52}\right) \cdot 17} \]

Calculate the growth rate: \[ \frac{\ln 13}{52} \approx 0.04933 \]

Substitute the growth rate and \( t = 17 \) into the formula: \[ y = 20 e^{0.04933 \cdot 17} \] \[ y \approx 20 e^{0.83861} \] \[ y \approx 20 \cdot 2.31299 \] \[ y \approx 46.2598 \]

Round the result to the nearest whole number: \[ y \approx 46 \]

Final Answer