a biologist begins working with a sample containing 20,000 bacteria. the population size doubles every 10…

a biologist begins working with a sample containing 20,000 bacteria. the population size doubles every 10 days according to the equation $p(t)=20,000(2)^{\frac{t}{10}}$, where $t$ represents the number of days. after seven days the biologist begins working with a second sample of 20,000 bacteria. the equation used to represent the population of sample 2 after $t$ days is $p(t)=20,000(2)^{\frac{t - 7}{10}}$. which of the following is an equivalent form of the equation for the population of sample 2?\n$p(t)=\frac{20,000cdot2^{t - 7}}{2^{10}}$\n$p(t)=\frac{20,000cdot2^{\frac{t}{10}}}{2^{7}}$\n$p(t)=\frac{20,000cdotsqrt7{2^{t}}}{2^{10}}$

a biologist begins working with a sample containing 20,000 bacteria. the population size doubles every 10 days according to the equation $p(t)=20,000(2)^{\frac{t}{10}}$, where $t$ represents the number of days. after seven days the biologist begins working with a second sample of 20,000 bacteria. the equation used to represent the population of sample 2 after $t$ days is $p(t)=20,000(2)^{\frac{t - 7}{10}}$. which of the following is an equivalent form of the equation for the population of sample 2?\n$p(t)=\frac{20,000cdot2^{t - 7}}{2^{10}}$\n$p(t)=\frac{20,000cdot2^{\frac{t}{10}}}{2^{7}}$\n$p(t)=\frac{20,000cdotsqrt7{2^{t}}}{2^{10}}$

Answer

Explanation:

Step1: Apply exponent - rule

Use the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$. Given $p(t)=20000(2)^{\frac{t - 7}{10}}$, we can rewrite $(2)^{\frac{t - 7}{10}}$ as $\frac{2^{\frac{t}{10}}}{2^{\frac{7}{10}}}$. But if we use the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$ on $2^{\frac{t - 7}{10}}$ in a different way, considering $a = 2$, $m=t - 7$ and $n = 10$, we have $2^{\frac{t - 7}{10}}=\frac{2^{\frac{t - 7}{1}}}{2^{10}}$. Also, using the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$ for $a = 2$, $m=t$ and $n = 7$ in the context of the original formula $p(t)=20000(2)^{\frac{t - 7}{10}}$, we know that $(2)^{\frac{t - 7}{10}}=\frac{2^{\frac{t}{10}}}{2^{\frac{7}{10}}}$. Another way is to use the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$ for the exponent part of the function. We know that $a^{m - n}=\frac{a^{m}}{a^{n}}$, so $(2)^{\frac{t - 7}{10}}=\frac{(2)^{\frac{t}{10}}}{(2)^{\frac{7}{10}}}$. If we rewrite the function $p(t)=20000(2)^{\frac{t - 7}{10}}$ using the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$ for the exponent of 2, we get $p(t)=\frac{20000\cdot2^{\frac{t}{10}}}{2^{\frac{7}{10}}}$. However, if we use the rule $a^{m - n}=\frac{a^{m}}{a^{n}}$ with $a = 2$, $m=t - 7$ and $n = 10$ in the form $2^{\frac{t - 7}{10}}=\frac{2^{t - 7}}{2^{10}}$, then $p(t)=\frac{20000\cdot2^{t - 7}}{2^{10}}$.

Step2: Conclusion

The equivalent form of the equation $p(t)=20000(2)^{\frac{t - 7}{10}}$ is $p(t)=\frac{20000\cdot2^{t - 7}}{2^{10}}$.

Answer:

$p(t)=\frac{20000\cdot2^{t - 7}}{2^{10}}$