Math  /  Algebra

QuestionA culture contains 15,000 bacteria, with the population increasing exponentially the culture contains 25,000 bacteria after 12 hours. a. Write a function in the form y=y0ekt y=y_{0} e^{\text {kt }} giving the number of bacteria after thours b. Write the function from part a in the form y=y0aty=y_{0} a^{t} c. How long will it be until there are 50,000 bacteria? a. The exponential equation is \square \square (Round to three decimal places as needed.)

Studdy Solution
To find the time tt when the number of bacteria is 50,000, use the function y=15000e0.033ty = 15000 e^{0.033t}.
Set yy to 50,000: 50000=15000e0.033t50000 = 15000 e^{0.033t}
Solve for tt: 5000015000=e0.033t\frac{50000}{15000} = e^{0.033t}
103=e0.033t\frac{10}{3} = e^{0.033t}
Take the natural logarithm of both sides: ln(103)=0.033t\ln\left(\frac{10}{3}\right) = 0.033t
t=ln(103)0.033t = \frac{\ln\left(\frac{10}{3}\right)}{0.033}
Calculate tt: t33.33 hourst \approx 33.33 \text{ hours}
The solution to the problem is: a. The exponential equation is y=15000e0.033ty = 15000 e^{0.033t}. b. In the form y=y0aty = y_0 a^t, the equation is y=150001.034ty = 15000 \cdot 1.034^t. c. It will take approximately 33.33 hours for the population to reach 50,000 bacteria.

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