Chapter 9.5, Problem 35E

(a)

To determine

The value of $u\cdot \left(v\times w\right)$.

Answer to Problem 35E

The value of $u\cdot \left(v\times w\right)$ is $27,00,000\sqrt{3}$.

Explanation of Solution

Given:

The fish tank is in the shape parallelepiped with base 300 cm long and 120 cm wide. The front and back arte vertical, left and right faces are slanted at $30°$ with sides 120 cm and 150 cm as shown in below figure,

Figure (1)

Formula used: The dot product of two vectors with magnitude $\left|u\right|$ and $\left|v\right|$ is given below,

$u\cdot v=\left|u\right|\left|v\right|\cos \theta$ (1)

Where,

• $\theta$ is the angle between the vectors u and v.

The cross product of two vectors with magnitude $\left|u\right|$ and $\left|v\right|$ is given below,

$\left|u\times v\right|=\left|u\right|\left|v\right|\sin \theta$ (2)

Where,

• $\theta$ is the angle between the vectors u and v.

Calculation:

The fish tank is in the shape parallelepiped with base 300 cm long and 120 cm wide, therefore the value of $\left|u\right|$ is 300 and $\left|w\right|$ is 120. The angle between u and w is $90°$.

The front and back arte vertical, left and right faces are slanted at $30°$ with sides 120 cm and 150 cm, therefore the value of $\left|v\right|$ is 150 and the angle between v and w is $60°$.

Substitute 150 for $\left|u\right|$, 120 for $\left|v\right|$ and $60°$ for $\theta$ in equation (2), to find the value of $\left|v\times w\right|$,

$\begin{array}{c}\left|v\times w\right|=\left(150\right)\left(120\right)\sin 60°\\ =\left(18000\right)\left(\frac{\sqrt{3}}{2}\right)\\ =9000\sqrt{3}\end{array}$

The vector $v\times w$ is the vector perpendicular to both the vectors v and w, therefore the vector $v\times w$ parallel to the vector u and the angle between u and $v\times w$ is 0.

From equation (1), the dot product of u and $v\times w$ is expressed as

$u\cdot \left(v\times w\right)=\left|u\right|\left|v\times w\right|\cos \theta$

Substitute 300 for $\left|u\right|$, $9000\sqrt{3}$ for $\left|v\times w\right|$ and 0 for $\theta$ in above equation, to find the value of $u\cdot \left(v\times w\right)$,

$\begin{array}{c}u\cdot \left(v\times w\right)=\left(300\right)\left(9000\sqrt{3}\right)\cos 0°\\ =27,00,000\sqrt{3}\left(1\right)\left(\because \cos 0°=1\right)\\ =27,00,000\sqrt{3}\end{array}$

Thus, the value of $u\cdot \left(v\times w\right)$ is $27,00,000\sqrt{3}$.

(b)

To determine

The capacity of tank in liters if $1\text{L}$ is equal to $1000{\text{cm}}^{\text{3}}$.

Answer to Problem 35E

The capacity of tank in liters if $1\text{L}$ is equal to $1000{\text{cm}}^{\text{3}}$ is $4677\text{litres}$.

Explanation of Solution

Given:

From part (a), the volume of the parallelepiped tank is $27,00,000\sqrt{3}{\text{cm}}^{\text{3}}$.

Calculation:

Divide the volume of parallelepiped tank by 1000 to find the capacity of tank in liters,

$\begin{array}{c}\text{Capacity}\text{of}\text{tank}\text{in}\text{liters}=\frac{27,00,000\sqrt{3}}{1000}\\ =2700\sqrt{3}\\ =4676.53718\\ =4677\end{array}$

Thus, the capacity of tank in liters if $1\text{L}$ is equal to $1000{\text{cm}}^{\text{3}}$ is $4677\text{litres}$.