Question

Rainfall Measurement Let the vector Rrepresent the amount of rainfall, in inches, whose direction is the inclination of the rain to a rain gauge. Let the vector Arepresent the area, in square inches, whose direction is the orientation of the opening of the rain gauge. See the figure. The volume of rain collected in the gauge, in cubic inches, is given by $V=\left|\mathbf{R}\mathbf{·}\mathbf{A}\right|,$even when the rain falls in a slanted direction or the gauge is not perfectly vertical.

Suppose that $\mathbf{R}=\left(0.75,-1.75\right)and\mathbf{A}=\left(0.3,1\right).$

(a) Find $\left|\left|\mathbf{R}\right|\right|$and $\left|\left|\mathbf{A}\right|\right|$and interpret the meaning of each.

(b) Compute Vand interpret its meaning.

(c) If the gauge is to collect the maximum volume of rain, what must be true about Rand A?

Short answer

Part a. The value $\left|\left|\mathbf{R}\right|\right|$is $1.9inch$which means that $1.9$inch of rainfall is received and the value $\left|\left|\mathbf{I}\right|\right|$is $1.04$square inches which means that the opening of the rain gauge is $1.04$square inches.

Part b. The value V is $1.525$cube inches, which means $1.525$cube inches of rainwater are collected by the rain gauge.

Part c. The maximum number of rain is $1.976$ cube inches when the vectors RandAare parallel.

Step-by-step solution

Part (a) Step 1. Given Information 

The amount of rainfall in the component form is $\left(0.75,-1.75\right)$and the area in the component form is $\left(0.3,1\right)$.

We have to find$\left|\left|\mathbf{R}\right|\right|and\left|\left|\mathbf{A}\right|\right|$and interpret the meaning of each.

Part (a) Step 2. Finding R and A

The vector Rcan be represented as

$\mathbf{R}=0.75\mathbf{i}-1.75\mathbf{j}$

So,

$$\begin{gathered} \left|\left|\mathbf{R}\right|\right|=\sqrt{{\left(0.75\right)}^2+{\left(-1.75\right)}^2} \\ \left|\left|\mathbf{R}\right|\right|=\sqrt{3.625} \\ \left|\left|\mathbf{R}\right|\right|\approx 1.9\mathrm{inches} \end{gathered}$$

It means that $1.9$inches amount of rainfall is received.

The vector Acan be represented as

$\mathbf{A}=0.3\mathbf{i}\mathbf{}+\mathbf{j}$

So,

$$\begin{gathered} \left|\left|\mathbf{A}\right|\right|=\sqrt{{\left(0.3\right)}^2+{\left(1\right)}^2} \\ \left|\left|\mathbf{A}\right|\right|=\sqrt{1.09} \\ \left|\left|\mathbf{A}\right|\right|=1.04{\mathrm{inch}}^2 \end{gathered}$$

It means that $1.04$square inches of the area are opening of the rain gauge.

Part (b) Step 1. Finding V

The V is given as $V=\left|\mathbf{R}·\mathbf{A}\right|$

So,

$$\begin{gathered} V=\left|\mathbf{R}·\mathbf{A}\right| \\ V=\left|\left(0.75\right)\left(0.3\right)+\left(-1.75\right)\left(1\right)\right| \\ V=\left|0.225-1.75\right| \\ V=1.525{\mathrm{inch}}^3 \end{gathered}$$

It means that$1.525$cube inches of rain water is collected by the rain gauge.

Part (c) Step 1. Finding the maximum volume of rain

The volume of rain collected by the rain gauge is $V=\left|\mathbf{R}·\mathbf{A}\right|$

Let $\theta ,0\le \theta \le \frac{\mathrm{\pi }}{2},$ be the angle between the two given vectors Rand A

So,

$$\begin{gathered} V=\left|\mathbf{R}·\mathbf{A}\right| \\ V=\left|\left|\left|\mathbf{R}\right|\right|\left|\left|\mathbf{A}\right|\right|\cos \theta \right| \\ V=\left|\left(1.9\right)\left(1.04\right)\cos \theta \right| \\ V=\left|1.976\cos \theta \right| \end{gathered}$$

The volume of the rain will be maximum when $\cos \theta$is maximum and $\cos \theta$is maximum at $\theta =0$

Thus, the maximum volume is$$\begin{gathered} V=\left|1.976\cos 0^{\circ }\right| \\ V=1.976{\mathrm{inch}}^3\left[\cos 0^{\circ }=1\right] \end{gathered}$$