Problem 31 - Algebra and Trigonometry

Question

Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun's rays. Let the vector \(\mathbf{A}\) represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by \(W=|\mathbf{I} \cdot \mathbf{A}|\) Suppose that \(\mathbf{I}=\langle-0.02,-0.01\rangle\) and \(\mathbf{A}=\langle 300,400\rangle\) (a) Find \(\|\mathbf{I}\|\) and \(\|\mathbf{A}\|,\) and interpret the meaning of each. (b) Compute \(W\) and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and \(\mathbf{A}\) ?

Step-by-Step Solution

Answer

Magnitudes: \(\|\mathbf{I}\| \approx 0.02236, \|\mathbf{A}\| = 500\). Collected energy: 10 watts. Maximum watts when \(\mathbf{I}\) and \(\mathbf{A}\) are parallel.

Step 1: Finding Magnitudes

First, calculate the magnitudes (norms) of the vectors \(\mathbf{I}\) and \(\mathbf{A}\). Use the magnitude formula \(\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\). For \(\mathbf{I} = \langle -0.02, -0.01 \rangle\) and \(\mathbf{A} = \langle 300, 400 \rangle\): \[ \|\mathbf{I}\| = \sqrt{(-0.02)^2 + (-0.01)^2} = \sqrt{0.0004 + 0.0001} = \sqrt{0.0005} \approx 0.02236 \] \[ \|\mathbf{A}\| = \sqrt{(300)^2 + (400)^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500 \]

Step 2: Interpreting Magnitudes

The magnitudes represent the intensity of the sunlight per square centimeter and the area of the solar panel in square centimeters, respectively. Thus, \(\|\mathbf{I}\| \) is approximately 0.02236 watts per square centimeter, and \(\|\mathbf{A}\|\) is 500 square centimeters.

Step 3: Computing the Dot Product

Calculate the dot product \(\mathbf{I} \cdot \mathbf{A}\) using the formula \(\mathbf{v} \cdot \mathbf{u} = v_1u_1 + v_2u_2\). For \(\mathbf{I} = \langle -0.02, -0.01 \rangle\) and \(\mathbf{A} = \langle 300, 400 \rangle\): \[ \mathbf{I} \cdot \mathbf{A} = (-0.02 \cdot 300) + (-0.01 \cdot 400) = -6 + (-4) = -10 \] Since watts are always positive, take the absolute value: \[ W = |\mathbf{I} \cdot \mathbf{A}| = |-10| = 10 \, \text{watts} \]

Step 4: Interpreting the Collected Watts

The total number of watts collected by the solar panel is 10 watts. This means that, given the current intensity and orientation, the panel will collect 10 watts of energy from the sun.

Step 5: Maximizing Collected Watts

To collect the maximum number of watts, the vectors \(\mathbf{I}\) and \(\mathbf{A}\) must be in the same direction, meaning they should be parallel. This can be achieved when \(\mathbf{I}\) and \(\mathbf{A}\) are either in the same direction (positive dot product) or directly opposed (still a positive magnitude after taking absolute value).

Key Concepts

Vector Magnitudes

When dealing with vectors, the magnitude (or norm) is a measure that describes how long or strong the vector is. It is calculated using the formula \(\rightarrow \sum ... \sqrt{v_1^2 + v_2^2} \). For example, to find the magnitude of vector \( \mathbf{I} = \langle -0.02, -0.01 \rangle \), we follow these steps:

\[ \|\mathbf{I}\| = \sqrt{(-0.02)^2 + (-0.01)^2} = \sqrt{0.0004 + 0.0001} = \sqrt{0.0005} \approx 0.02236 \]
This tells us that the sunlight intensity is approximately 0.02236 watts per square centimeter. In the same way, the magnitude of vector \( \mathbf{A} \) signifies the area of the solar panel and is calculated as:

\[\|\mathbf{A}\| = \sqrt{(300)^2 + (400)^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500 \]
This shows that the area of the panel is 500 square centimeters. Magnitudes give us crucial information on the scale or strength of vectors.

Solar Energy Collection

The collection of solar energy by a panel depends on two main factors: the intensity of the sunlight and the surface area of the panel that captures this energy. These are represented by vectors \(\mathbf{I}\) and \(\mathbf{A}\) respectively. The formula for the total watts collected is given by the dot product of these vectors:

\( W = |\mathbf{I} \cdot \mathbf{A}| \)
Calculating the dot product involves multiplying the corresponding components of the vectors and summing the results:

\[ \mathbf{I} \cdot \mathbf{A} = (-0.02 \cdot 300) + (-0.01 \cdot 400) = -6 + (-4) = -10 \]
Taking the absolute value gives us:

\( W = | -10 | = 10 \text{ watts} \)
This means that with the given sunlight intensity and panel area, the solar panel can collect 10 watts of energy.

Vector Orientation

The direction or orientation of vectors \( \mathbf{I} \) and \( \mathbf{A} \) plays a crucial role in the efficiency of energy collection. If the vectors are aligned perfectly, meaning they are either pointing in the same direction or exactly opposite each other, the collection of energy is maximized because the dot product reaches its highest (in the absolute value). For maximum watts:

• The vectors should be parallel and in the same direction.
  
• If not, vectors could be anti-parallel (directly opposed) but still produce maximum magnitude after absolute value.
  

When vectors are aligned, the angle between them is either 0 or 180 degrees, and this provides the most efficient energy transfer through maximum dot product calculation.