Chapter 1.6, Problem 45E

To determine

To find: How many miles to the right of the origin need to be moved before the height of the curve reaches 3 ft if the graph $y={\log }_{2}x$ is drawn on the coordinate plane with the scale, one unit equals one inch.

Answer to Problem 45E

The distance has to be moved to the right of the origin is 1084587.7 miles.

Explanation of Solution

The given function is, $y={\log }_{2}x$.

Since 1 ft = 12 inches, the curve reaches 3 ft which is equivalent to 36 inches.

So, substitute $y=36$ in the given function and then solve for x.

$\begin{array}{l}36={\log }_{2}x\\ 2^{36}=2^{{\log }_{2}x}\left[\because a^{{\log }_{a}x}=x\right]\\ 2^{36}=x\\ x=68719476736\end{array}$

Thus, $x=68719476736$ inches.

Since 1 mile = 63360 inches, $x\approx 1084587.7$ miles.

Hence, the distance has to be moved to the right of the origin is 1084587.7 miles.