Chapter 72, Problem 2A

To determine

The angle $\angle x$ to the nearest minute.

The dimension $y$ to three decimal places.

Answer to Problem 2A

The angle $\angle x$ to the nearest minute is $=46°6\text{'}$.

The dimension $y$ to three decimal places is $\text{y=AE}=85.788\text{ mm}$

Explanation of Solution

Given:

  

A figure as above, in which different dimensions are shown.

  $\text{AB}=77.02\text{3 mm}$, $\text{BC}=\text{CD}=\text{DE}=\text{21}\text{.755 mm}$

Concept Used:

For a right angle triangle

$\text{tanθ=}\frac{\text{perpendicular}}{\text{base}}$

  $\begin{array}{l}\text{ sinθ=}\frac{\text{Perpendicular}}{\text{hypotenuse}}\\ \text{hypotenuse=}\frac{\text{Perpendicular}}{\text{sinθ}}\end{array}$

Calculation:

As per the given problem in the $\text{ΔABC}$

  $\text{Perpendicular, BC}=\text{21}\text{.755 mm}$, $\text{Base, AB}=77.02\text{3 mm}$.

Substitute these values in the formula $\text{tanθ=}\frac{\text{perpendicular}}{\text{base}}$, we get

When $\theta =\angle a$

  $\begin{array}{c}\text{tan}\angle a\text{=}\frac{\text{21}\text{.755}}{\text{77}\text{.023}}\\ \angle a{\text{=tan}}^{-1}\frac{\text{21}\text{.755}}{\text{77}\text{.023}}\\ \angle a\text{=15}\text{.77}°\end{array}$

The hypotenuse of the $\text{ΔABC}$ is $\text{AC}$

  $\begin{array}{c}\text{hypotenuse=}\frac{\text{Perpendicular}}{\text{sin}\angle a}\\ \text{AC=}\frac{\text{Perpendicular}}{\text{sin}\angle a}\end{array}$

Substitute values in the formula $\text{AC=}\frac{\text{Perpendicular}}{\text{sin}\angle a}$, we get

  $\begin{array}{l}\text{AC}=\frac{\text{21}\text{.755 mm}}{\text{sin15}\text{.77}°}\\ \text{AC}=80.0\text{5 mm}\end{array}$

Now in the $\text{ΔACD}$

  $\begin{array}{c}\text{tan}\angle b\text{=}\frac{\text{21}\text{.755}}{\text{80}\text{.05}}\\ \angle b{\text{=tan}}^{-1}\frac{\text{21}\text{.755}}{\text{80}\text{.05}}\\ \angle b\text{=}15.2°\end{array}$

The hypotenuse of the $\text{ΔACD}$ is $\text{AD}$

Substitute values in the formula $\text{AD=}\frac{\text{CD}}{\text{sin}\angle b}$

  $\begin{array}{l}\text{AD}=\frac{\text{21}\text{.755 mm}}{\text{sin15}.2°}\\ \text{AD}=82.97\text{ mm}\end{array}$

Now in the $\text{ΔADE}$

  $\begin{array}{c}\text{tan}\angle c\text{=}\frac{\text{DE}}{\text{AD}}\\ \angle c{\text{=tan}}^{-1}\frac{\text{21}\text{.755}}{\text{82}\text{.97}}\\ \angle c\text{=}14.69°\end{array}$

The hypotenuse of the $\text{ΔADE}$ is $\text{AE}=y$

Substitute values in the formula $\text{AE}=y\text{=}\frac{\text{DE}}{\text{sin}\angle c}$

  $\begin{array}{c}\text{AE}=\frac{\text{21}\text{.755 mm}}{\text{sin14}.69°}\\ \text{y=AE}=85.788\text{ mm}\end{array}$

The dimension $y$ to three decimal places is $\text{y=AE}=85.788\text{ mm}$

The angle $\angle x=\angle a+\angle b+\angle c$

  $\begin{array}{l}\angle x=15.77°+15.2°+14.69°\\ \angle x=15.77°+15.2°+14.69°\\ \angle x=45.66°\\ \angle x=46°6\text{'}\end{array}$

The angle $\angle x$ to the nearest minute is $=46°6\text{'}$.

Conclusion:

Therefore, the angle $\angle x$ to the nearest minute is $=46°6\text{'}$.

Therefore, the dimension $y$ to three decimal places is $\text{y=AE}=85.788\text{ mm}$.