When arriving at the final answer for a double inequality question, it appears that my text book has truncated one part and rounded the other. Is there some weird inequality rule that I don't know about?
The text book gives the answer for a safe angle to lean a ladder to the right against a wall as:
$71.5 ^\circ \leq \text{safe angle} \leq 76.0 ^\circ \space (1dp)$
The numbers generated by my calculator (see the calculation below) for this problem are:
71.56505. . . and 75.96375. . .
As you can see, the lower number 71.5 has been truncated at (1dp), but the higher number 76.0 has been rounded at (1dp). Why was the lower number truncated?
The original question is:
If the gradient of the ladder is m, the safe angle at the base to lean against a wall is expressed as an inequality:
$ 3 \leq \text{m} \leq 4 $
Find the angles at the base to complete the following inequality:
$ \theta_1 \leq \text {safe angle} \leq \theta_2 $
I worked out the angles, using the gradients above, by thinking of the ladder as making an angle with the positive x-axis, as below:
$$\begin{align}
\tan \theta_1 & = 3 \\
\theta_1 & = 71.6 ^\circ \space \space (1dp ) \\
\end{align}
$$
and
$$\begin{align}
\tan\theta_2 & = 4 \\
\theta_2 & = 76.0 ^\circ \space \space (1dp) \\
\end{align}$$
So I put my answer as:
$71.6 ^\circ \leq \text{safe angle} \leq 76.0 ^\circ \space (1dp)$
which has the first number out by 0.1$^\circ$ when rounding both numbers.
Cheers for any help!