Dr. Schmidt recorded a count of 100 patients infected with a disease After 15 weeks, they counted 212 patients. Assuming the count of patients infected grows at a continuous rate, write an exponential model for this situation in terms of t.the number of weeks passed. Round to two decimal places.when necessary. Show your work here Hint: To add an exponent (x^y) type "exponent" or press "A" I A(t)=

To write an exponential model for the situation, we can use the formula for continuous growth:<br /><br />\[ A(t) = A_0 \cdot e^{rt} \]<br /><br />where:<br />- \( A(t) \) is the amount at time \( t \),<br />- \( A_0 \) is the initial amount,<br />- \( r \) is the continuous growth rate,<br />- \( t \) is the time in weeks,<br />- \( e \) is the base of the natural logarithm.<br /><br />Given:<br />- Initial count \( A_0 = 100 \),<br />- Count after 15 weeks \( A(15) = 212 \).<br /><br />We need to find the growth rate \( r \). Using the information provided:<br /><br />\[ 212 = 100 \cdot e^{15r} \]<br /><br />First, solve for \( e^{15r} \):<br /><br />\[ e^{15r} = \frac{212}{100} = 2.12 \]<br /><br />Now, take the natural logarithm of both sides to solve for \( r \):<br /><br />\[ \ln(e^{15r}) = \ln(2.12) \]<br /><br />\[ 15r = \ln(2.12) \]<br /><br />\[ r = \frac{\ln(2.12)}{15} \]<br /><br />Calculate \( r \):<br /><br />\[ r \approx \frac{0.7514}{15} \approx 0.0501 \]<br /><br />Now, substitute \( A_0 \) and \( r \) back into the exponential model:<br /><br />\[ A(t) = 100 \cdot e^{0.0501t} \]<br /><br />So, the exponential model is:<br /><br />\[ A(t) = 100 \cdot e^{0.0501t} \]