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:

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

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 ${}_{\omega }A=\langle 0.3,1\rangle$ . The volume of rain collected in

the gauge, in cubic inches, is given by $=|R\cdot A|,$ 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 $A=0.3i+j$

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

Therefore the amount of rain $\text{fall}0.022{\text{in}}^2$ ,

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

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

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

Therefore the amount of rain $\text{fall}0.022{\text{in}}^2$ ,

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

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

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 $1.525{\text{in}}^3$ .

Explanation of Solution

Given:

Let the vector $R$ represent the amount of rainfall, in inches, $R=\langle 0.75,-1.75$ ) and let the

vector A represent the area, in square inches, $A=\langle 0.3,1$ ). The volume of rain collected in

the gauge, in cubic inches, is given $\text{by},=|R\cdot A|,$ even when the rain falls in a slanted

direction or the gauge is not perfectly vertical.

Calculation:

  $\text{b}\cdot V=|R\cdot A|=|0.75(0.3)-1.75(1)|=1.525$

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

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 ${}_{\alpha }A=\langle 0.3,1$ ). The volume of rain collected in

the gauge, in cubic inches, is given by $=|R\cdot A|$ , even when the rain falls in a slanted

direction or the gauge is not perfectly vertical.

Calculation:

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

  $\theta ={\cos }^{-1}(-0.77)={140.35}^{°}.$

Therefore the vectors $R$ many $A$ neither parallel nor. Orthogonal.