Question

A new kind of rocket takes off along an exponential trajectory, with height, in miles, represented by 3x, where x is the time, in seconds. Find the time when the height of the rocket is 8 miles.

What is the exact solution written as a logarithm?

What is an approximate solution rounded to the nearest thousandth?

Answer 1

We have been given that a new kind of rocket takes off along an exponential trajectory, with height, in miles, represented by $3^x$, where x is the time, in seconds. We are asked to find the time when the height of the rocket id 8 miles.

To find the time, we will equate height function with 8 and solve for x as:

$3^x=8$

To solve for x, we will take natural log on both sides as:

$\ln(3^x)=\ln (8)$

We can rewrite 8 as $2^3$.

$\ln(3^x)=\ln (2^3)$

Using natural log property $\ln(a^b)=b\cdot \ln(a)$, we will get:

$x\cdot \ln(3)=3\cdot\ln (2)$

$\frac{x\cdot \ln(3)}{ \ln(3)}=\frac{3\cdot\ln (2)}{ \ln(3)}$

$x=\frac{3\cdot\ln (2)}{ \ln(3)}$

Therefore, exact solution will be $\frac{3\cdot\ln (2)}{ \ln(3)}$.

$x=\frac{2.0794415416798359}{1.0986122886681097}$

$x=1.892789260714$

Upon rounding to nearest thousandth, we will get:

$x\approx 1.89$

Therefore, the height of the rocket will be 8 miles after approximately 1.89 seconds.