Chapter 9.5, Problem 28AYU

To determine

To find:

a. Find $\Vert R\Vert$ and $\Vert A\Vert$, and interpret the meaning of each.

Answer to Problem 28AYU

Solution:

a. The amount of rain fall $0.022i{n}^2$, The area of the opening of rain gauge is $1.04i{n}^2$.

Explanation of Solution

Given:

Let the vector $R$ represent the amount of rainfall, in inches, $R=\langle 0.75,-1.75\rangle$ and let the vector A represent the area, in square inches, $A=\langle 0.3,1\rangle$. The volume of rain collected in the gauge, in cubic inches, is given by $=\left|R\cdot A\right|$, even when the rain falls in a slanted direction or the gauge is not perfectly vertical.

Calculation:

a. If $\langle a,b\rangle$ denotes the position vector of a vector $v$, then the vector $v$ is expressed as,

$\langle a,b\rangle =a\langle 1,0\rangle +b\langle 0,1\rangle =ai+bj$

Thus the vector $v$ is $\langle 0.75,-1.75\rangle =a\langle 0.75,0\rangle +b\langle 0,-1.75\rangle =0.75i-1.75j$,

$R=0.75i-1.75j$

Now the area of the vector $A$ is,

$\langle 0.3,1\rangle =a\langle 0.3,0\rangle +b\langle 0,1\rangle =0.3i+j$

Therefore the vector $A$ is $A=0.3i+j$,

$\Vert R\Vert =\sqrt{{\left(0.75\right)}^2+{\left(-1.75\right)}^2}=\sqrt{3.625}=1.90$

Therefore the amount of rain fall $0.022i{n}^2$,

$\Vert A\Vert =\sqrt{{\left(0.3\right)}^2+{\left(1\right)}^2}=1.04$

The area of the opening of rain gauge is $1.04i{n}^2$.

To determine

To find:

b. Compute $V$ and interpret its meaning.

Answer to Problem 28AYU

Solution:

b. The volume of rain collected in the gauge is $1.525i{n}^3$.

Explanation of Solution

Given:

Let the vector $R$ represent the amount of rainfall, in inches, $R=\langle 0.75,-1.75\rangle$ and let the vector A represent the area, in square inches, $A=\langle 0.3,1\rangle$. The volume of rain collected in the gauge, in cubic inches, is given by $=\left|R\cdot A\right|$, even when the rain falls in a slanted direction or the gauge is not perfectly vertical.

Calculation:

b. $V=\left|R\cdot A\right|=\left|0.75\left(0.3\right)-1.75\left(1\right)\right|=1.525$

The volume of rain collected in the gauge is $1.525i{n}^3$.

To determine

To find:

c. If the gauge is to collect the maximum volume of rain, what must be true about $R$ and $A$?

Answer to Problem 28AYU

Solution:

c. The vectors $R$ and $A$ neither parallel nor orthogonal.

Explanation of Solution

Given:

Let the vector $R$ represent the amount of rainfall, in inches, $R=\langle 0.75,-1.75\rangle$ and let the vector A represent the area, in square inches, $A=\langle 0.3,1\rangle$. The volume of rain collected in the gauge, in cubic inches, is given by $=\left|R\cdot A\right|$, even when the rain falls in a slanted direction or the gauge is not perfectly vertical.

Calculation:

c. $\cos \theta =\frac{R\cdot A}{\Vert R\Vert \Vert A\Vert }=\frac{-1.525}{1.9\left(1.04\right)}=-0.77$

$\theta ={\cos }^{-1}\left(-0.77\right)={140.35}^{\circ }$. Therefore the vectors $R$ and $A$ neither parallel nor orthogonal.