Chapter 1, Problem 73AP

(a)

To determine

The distance between the cow A and cow B.

Answer to Problem 73AP

The distance between the cow A and cow B is $12.1\text{m}$.

Explanation of Solution

The figure 1 below shows the location of the cow A, cow B and the person.

Use the law of cosine to write the formula for the sides a, b, and c.

    $a^2=b^2+c^2-2bc\cos A$

    $b^2=c^2+a^2-2ca\cos B$

    $c^2=a^2+b^2-2ab\cos C$

The distance between the cows is the side $c$.

It is given that, $a=25.0\text{m}$, $b=15.0\text{m}$, and angle $C=20.0°$.

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, $20.0°$ for $C$ to determine the side $c$.

    $\begin{array}{c}c=\sqrt{{\left(25.0\text{m}\right)}^2+{\left(15.0\text{m}\right)}^2-2\left(25.0\text{m}\right)\left(15.0\text{m}\right)\cos 20.0°}\\ =12.1\text{m}\end{array}$

The distance between the cow A and cow B is $12.1\text{m}$.

(b)

To determine

The angle between the person and cow B as seen by the cow A.

Answer to Problem 73AP

The angle between the person and cow B as seen by the cow A is $135°$.

Explanation of Solution

The figure 1 below shows the location of the cow A, cow B and the person.

Use the law of cosine to write the formula for the sides a, b, and c.

    $a^2=b^2+c^2-2bc\cos A$

    $b^2=c^2+a^2-2ca\cos B$

    $c^2=a^2+b^2-2ab\cos C$

The angle between the person and cow B as seen by cow A is the angle $A$.

Re-write the above equation to get an expression for $A$.

    $A={\cos }^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)$

It is given that, $a=25.0\text{m}$, $b=15.0\text{m}$, and angle $C=20.0°$. From section (a) $c=12.1\text{m}$.

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, $12.1\text{m}$ for $c$ to determine the side $A$.

    $\begin{array}{c}A={\cos }^{-1}\left(\frac{{\text{(25}\text{.0}\text{m)}}^{\text{2}}-{\text{(15}\text{.0}\text{m)}}^{\text{2}}-{\text{(12}\text{.1}\text{m)}}^{\text{2}}}{\text{2(15}\text{.0}\text{m)(12}\text{.1}\text{m)}}\right)\\ =134.8°\\ \approx 135°\end{array}$

The angle between the person and cow B as seen by the cow A is $135°$.

(c)

To determine

The angle between the person and cow A as seen by the cow B.

Answer to Problem 73AP

The angle between the person and cow A as seen by the cow B is $25.2°$.

Explanation of Solution

The figure 1 below shows the location of the cow A, cow B and the person.

Use the law of cosine to write the formula for the sides a, b, and c.

    $a^2=b^2+c^2-2bc\cos A$

    $b^2=c^2+a^2-2ca\cos B$

    $c^2=a^2+b^2-2ab\cos C$

The angle between the person and cow A as seen by cow B is the angle $B$.

Re-write the above equation to get an expression for $B$.

    $B={\cos }^{-1}\left(\frac{c^2+a^2-b^2}{2ca}\right)$

It is given that, $a=25.0\text{m}$, $b=15.0\text{m}$, and angle $C=20.0°$. From section (a) $c=12.1\text{m}$.

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, $12.1\text{m}$ for $c$ to determine the side $B$.

    $\begin{array}{c}B={\cos }^{-1}\left(\frac{{\text{(15}\text{.0}\text{m)}}^{\text{2}}-{\text{(25}\text{.0}\text{m)}}^{\text{2}}-{\text{(12}\text{.1}\text{m)}}^{\text{2}}}{\text{2(25}\text{.0}\text{m)(12}\text{.1}\text{m)}}\right)\\ =25.2°\end{array}$

The angle between the person and cow A as seen by the cow B is $25.2°$.

(d)

To determine

The angle between star B and sun as observed by an inhabitant of a planet orbiting star A.

Answer to Problem 73AP

The angle between star B and sun as observed by an inhabitant of a planet orbiting star A is $135°$.

Explanation of Solution

The figure 2 below shows the location of the star A, star B and the sun S.

Use the law of cosine to write the formula for the sides a, b, and c.

    $a^2=b^2+c^2-2bc\cos A$

    $b^2=c^2+a^2-2ca\cos B$

    $c^2=a^2+b^2-2ab\cos C$

The angle between the person and cow A as seen by cow B is the angle $B$.

Re-write the above equation to get an expression for $A$.

    $A={\cos }^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)$

It is given that, $a=25.0\text{ly}$, $b=15.0\text{ly}$, and angle $C=20.0°$.

Conclusion:

Substitute $25.0\text{ly}$ for $a$, $15.0\text{ly}$ for $b$, $12.1\text{m}$ for $c$ to determine the side $A$.

    $\begin{array}{c}A={\cos }^{-1}\left(\frac{{\text{(25}\text{.0}\text{ly)}}^{\text{2}}-{\text{(15}\text{.0}\text{ly)}}^{\text{2}}-{\text{(12}\text{.1}\text{ly)}}^{\text{2}}}{\text{2(15}\text{.0}\text{ly)(12}\text{.1}\text{ly)}}\right)\\ =134.8°\\ \approx 135°\end{array}$

The angle between star B and sun as observed by an inhabitant of a planet orbiting star A is $135°$.