Chapter 1, Problem 70P

(a)

To determine

The distance between the cow A and cow B.

Answer to Problem 70P

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

Explanation of Solution

The figure 1 shows the geometry of the problem says to use the law of cosines to known sides and angles of a triangle to the unknown sides and angle.

Here, $a,b,\text{ and }c$ are the sides of the triangle with opposite angles $A,B,\text{ and }C$

The laws of cosines are,

  $\begin{array}{c}{a}^2=b^2+c^2-2bc\cos A\\ b^2=c^2+a^2-2ca\cos B\\ c^2=a^2+b^2-2ab\cos C\end{array}$

The cows in the meadow region, the triangle has sides $a=25.0\text{m}$ and $b=15.0\text{m}$, angle $C=20.0°$.

Here, the object A is cow A, object B is cow B, and object C is person.

From the law of cosine, find the side $c$.

  $c^2=a^2+b^2-2ab\cos C$        (I)

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, and $20.0°$ for $C$ in equation (I) to find $c$.

  $\begin{array}{c}{c}^2={\left(25.0\text{m}\right)}^2+{\left(15.0\text{m}\right)}^2-2\left(25.0\text{m}\right)\left(15.0\text{m}\right)\cos \left(20.0°\right)\\ =145.23{\text{m}}^2\\ c=12.1\text{m}\end{array}$

Therefore, 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.

Answer to Problem 70P

The angle between the person and cow B is $134°$.

Explanation of Solution

From part (a) use the law of cosine and find the angle $A$.

  $a^2=b^2+c^2-2bc\cos A$        (II)

Rewrite the above equation for $\cos A$.

  $\cos A=\frac{a^2-b^2-c^2}{2bc}$

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, and $12.1\text{m}$ for $c$ in above relation to find angle $A$.

  $\begin{array}{c}\cos A=\frac{{\left(25.0\text{m}\right)}^2-{\left(15.0\text{m}\right)}^2-{\left(12.1\text{m}\right)}^2}{2\left(15.0\text{m}\right)\left(12.1\text{m}\right)}\\ =0.6986\\ A={\cos }^{-1}\left(0.6986\right)\\ =180°-45.7°\\ =134°\end{array}$

Therefore, the angle between the person and cow B is $134°$.

(c)

To determine

The angle between the person and cow A.

Answer to Problem 70P

The angle between the person and cow A is $25.4°$.

Explanation of Solution

From part (a) use the law of cosine and find the angle $B$.

  $b^2=c^2+a^2-2ca\cos B$        (II)

Rewrite the above equation for $\cos B$.

  $\cos B=\frac{b^2-c^2-a^2}{2ca}$

Conclusion:

Substitute $25.0\text{m}$ for $a$, $15.0\text{m}$ for $b$, and $12.1\text{m}$ for $c$ in above relation to find angle $B$.

  $\begin{array}{c}\cos B=\frac{{\left(15.0\text{m}\right)}^2-{\left(12.1\text{m}\right)}^2-{\left(25.0\text{m}\right)}^2}{2\left(12.1\text{m}\right)\left(25.0\text{m}\right)}\\ =-0.9032\\ B={\cos }^{-1}\left(-0.9032\right)\\ =180°-154.6°\\ =25.4°\end{array}$

Therefore, the angle between the person and cow A is $25.4°$.

(d)

To determine

The angle between star B and Sun.

Answer to Problem 70P

The angle between star B and Sun is $134°$.

Explanation of Solution

Here, the situation is, object A is considered as star A, object B is star B, and object C is our Sun, so the triangle has sides $a=25.0\text{ly}$ and $b=15.0\text{ly}$, angle $C=20.0°$.

Conclusion:

Refer part (b),

  $\cos A=\frac{a^2-b^2-c^2}{2bc}$

Substitute $25.0\text{ly}$ for $a$, $15.0\text{ly}$ for $b$, and $12.1\text{ly}$ for $c$ in above relation to find angle $A$.

  $\begin{array}{c}\cos A=\frac{{\left(25.0\text{ly}\right)}^2-{\left(15.0\text{ly}\right)}^2-{\left(12.1\text{ly}\right)}^2}{2\left(15.0\text{ly}\right)\left(12.1\text{ly}\right)}\\ =0.6986\\ A={\cos }^{-1}\left(0.6986\right)\\ =180°-45.7°\\ =134°\end{array}$

Therefore, the angle between star B and Sun is $134°$.