Problem 76 - Algebra 2

Question

ENERGY A circular cell must deliver 18 watts of energy. If each square centimeter of the cell that is in sunlight produces 0.01 watt of energy, how long must the radius of the cell be?

Step-by-Step Solution

Answer

The radius of the cell must be approximately 23.93 cm.

Step 1: Understanding the Problem

We are given that a circular solar cell needs to deliver 18 watts of energy. Each square centimeter of the cell in sunlight generates 0.01 watt. We need to find out the radius that will achieve this output.

Step 2: Calculate Required Area

Since each square centimeter produces 0.01 watt and we need 18 watts, we calculate the total area required by dividing the total energy needed by the energy produced per square centimeter: \[ \text{Required Area} = \frac{18 \text{ watts}}{0.01 \text{ watts/cm}^2} = 1800 \text{ cm}^2. \]

Step 3: Area Formula for Circle

The area of a circle is given by the formula: \[ A = \pi r^2. \] Here, we have the area as 1800 cm², and need to find the radius \( r \).

Step 4: Solve for Radius

Set the area formula equal to the required area and solve for \( r \): \[ \pi r^2 = 1800. \] Divide both sides by \( \pi \): \[ r^2 = \frac{1800}{\pi}. \] Then, take the square root of both sides to find \( r \): \[ r = \sqrt{\frac{1800}{\pi}}. \]

Step 5: Numerical Calculation

Calculate the numerical value: \[ r = \sqrt{\frac{1800}{3.14159}} \approx \sqrt{573.026} \approx 23.93 \text{ cm}. \]

Key Concepts

Energy calculation

In the world of solar cells, energy calculation plays a crucial role in determining the performance and requirements of the cells. Solar cells convert sunlight into electricity, measured in watts. But how exactly do you calculate the energy required for a solar cell to function efficiently? Let's break it down.

In this problem, we have a circular solar cell that must produce 18 watts. Each part of the solar cell, specifically each square centimeter, generates only 0.01 watt when exposed to sunlight. To find how many square centimeters are necessary to reach the 18 watts target, you simply divide the total desired energy by the energy production per unit area.

This results in the formula:

• \[ \text{Required Area} = \frac{\text{Total Energy Required}}{\text{Energy Produced per Unit Area}} \]

Which gives us:

• \[ \text{Required Area} = \frac{18\, \text{watts}}{0.01\, \text{watts/cm}^2} = 1800\, \text{cm}^2 \]

Knowing the required area allows us to delve into the properties of circles to determine further dimensions of the solar cell.

Circle area formula

Once you know the required area of a solar cell, it's essential to relate this area to the shape of the cell. In our exercise, we are dealing with a circular solar cell. Thus, understanding the circle area formula is key.

The formula for the area of a circle is denoted as:

• \[ A = \pi r^2 \]

Where:

• \( A \) is the area
• \( r \) is the radius
• \( \pi \) is a constant approximately equal to 3.14159

In this scenario, we know that the area \( A \) of the circular solar cell needs to be 1800 cm² to produce the desired amount of energy, 18 watts.

The next step is to use this formula to determine the radius, which leads us to the following calculation:

• \[ \pi r^2 = 1800 \]

This equation becomes the basis for finding the correct radius of the solar cell.

Radius determination

Finding the radius of the solar cell involves solving the circle's area formula for \( r \). When you rearrange the equation \( \pi r^2 = 1800 \), you first need to isolate \( r^2 \):

• \[ r^2 = \frac{1800}{\pi} \]

This means you divide the area by \( \pi \) to get \( r^2 \). Now, to find the radius itself, you take the square root of both sides of the equation:

• \[ r = \sqrt{\frac{1800}{\pi}} \]

Carrying out this computation gives the numerical value for the radius. In this case:

• \[ r = \sqrt{\frac{1800}{3.14159}} \approx \sqrt{573.026} \approx 23.93 \text{ cm} \]

This result tells us the required radius of the solar cell. A perfect circle with this radius will have the needed area to deliver precisely 18 watts of power. Understanding this process is essential for designing solar cells that meet specific energy requirements.