Question

precalculus exponential, logistic & logarithmic functions form b
name mshammad awan
calculator is allowed on this portion of the test!

1. use a calculator to solve: $2^{3x}=165$

a 2.392 b 2.455 c 28.537 d 4.823

1. calculate to the nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded continuously at 3.4% using the formula $a = pe^{rt}$, where $p$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years.

a 2.2 years b 11.7 years c 14.5 years d 20.4 years

1. if there are initially 2100 bacteria in a culture, and the number of bacteria triple each hours, the number of bacteria after $t$ hours can be found using the formula $y = 2100(3)^{t}$. how long will it take the culture to grow to 50,000 bacteria?

a 2.89 hours b 4.15 hours c 1.47 hours d 5.01 hours

Explanation:

8.

Step1: Take the natural - log of both sides

Take the natural - log of $2^{3x}=165$. Using the property $\ln(a^b)=b\ln(a)$, we get $3x\ln(2)=\ln(165)$.

Step2: Solve for $x$

$x = \frac{\ln(165)}{3\ln(2)}$. Using a calculator, $\ln(165)\approx5.1003$ and $\ln(2)\approx0.6931$. Then $x=\frac{5.1003}{3\times0.6931}=\frac{5.1003}{2.0793}\approx2.455$.

Step1: Set up the equation for doubling

If the money doubles, $A = 2P$. Substitute into the formula $A = Pe^{rt}$: $2P=Pe^{0.034t}$.

Step2: Simplify the equation

Divide both sides by $P$ (since $P
eq0$), we get $2 = e^{0.034t}$.

Step3: Take the natural - log of both sides

$\ln(2)=\ln(e^{0.034t})$. Using the property $\ln(e^a)=a$, we have $\ln(2)=0.034t$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{0.034}$. Since $\ln(2)\approx0.6931$, then $t=\frac{0.6931}{0.034}\approx20.4$ years.

Step1: Set up the equation

Set $y = 50000$ in the formula $y = 2100(3)^t$. So, $50000=2100(3)^t$.

Step2: Isolate the exponential term

Divide both sides by 2100: $\frac{50000}{2100}=3^t$, which simplifies to $\frac{500}{21}=3^t$.

Step3: Take the natural - log of both sides

$\ln(\frac{500}{21})=\ln(3^t)$. Using the property $\ln(a^b)=b\ln(a)$ and $\ln(\frac{a}{b})=\ln(a)-\ln(b)$, we have $\ln(500)-\ln(21)=t\ln(3)$.

Step4: Solve for $t$

$\ln(500)\approx6.2146$, $\ln(21)\approx3.0445$, $\ln(3)\approx1.0986$. Then $t=\frac{\ln(500)-\ln(21)}{\ln(3)}=\frac{6.2146 - 3.0445}{1.0986}=\frac{3.1701}{1.0986}\approx2.89$ hours.

Answer:

B. 2.455

9.