The Total Area Of Two Square Windows Is $1,025 , \text{in}^2$. Each Side Of The Larger Window Is 5 Inches Longer Than The Sides Of The Smaller Window. How Long Are The Sides Of The Smaller Window? A = S 2 A = S^2 A = S 2 A. 13.5 In.B. 20 In.C. 25

Introduction

In this article, we will delve into a mathematical problem involving two square windows. The total area of the two windows is given as $1,025 \, \text{in}^2$, and we are asked to find the length of the sides of the smaller window. The problem states that each side of the larger window is 5 inches longer than the sides of the smaller window. We will use algebraic equations to solve this problem and find the length of the sides of the smaller window.

Understanding the Problem

Let's denote the length of the side of the smaller window as $s$. Since the larger window has sides that are 5 inches longer than the smaller window, the length of the side of the larger window is $s + 5$. The area of a square is given by the formula $A = s^2$, where $A$ is the area and $s$ is the length of the side.

Setting Up the Equation

We are given that the total area of the two windows is $1,025 \, \text{in}^2$. The area of the smaller window is $s^2$, and the area of the larger window is $(s + 5)^2$. Therefore, we can set up the equation:

$$s^2 + (s + 5)^2 = 1025$$

Expanding the Equation

To simplify the equation, we can expand the squared term:

$$s^2 + s^2 + 10s + 25 = 1025$$

Combine like terms:

$$2s^2 + 10s + 25 = 1025$$

Rearranging the Equation

Subtract 1025 from both sides of the equation:

$$2s^2 + 10s - 1000 = 0$$

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

$$s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 2$, $b = 10$, and $c = -1000$. Plugging these values into the formula, we get:

$$s = \frac{-10 \pm \sqrt{100 + 8000}}{4}$$

Simplify the expression under the square root:

$$s = \frac{-10 \pm \sqrt{8100}}{4}$$

$$s = \frac{-10 \pm 90}{4}$$

Finding the Length of the Sides

We have two possible solutions for $s$:

$$s = \frac{-10 + 90}{4} = 20$$

$$s = \frac{-10 - 90}{4} = -25$$

Since the length of the side of a window cannot be negative, we discard the solution $s = -25$. Therefore, the length of the sides of the smaller window is $s = 20$ inches.

Conclusion

In this article, we solved a mathematical problem involving two square windows. We used algebraic equations to find the length of the sides of the smaller window, given that the total area of the two windows is $1,025 \, \text{in}^2$ and each side of the larger window is 5 inches longer than the sides of the smaller window. The length of the sides of the smaller window is $s = 20$ inches.

Discussion

What do you think about this problem? Do you have any questions or comments? Share your thoughts in the discussion section below.

References

• [1] Algebraic equations, Wikipedia.
• [2] Quadratic formula, Wikipedia.

Related Articles

• [1] Solving quadratic equations using the quadratic formula.
• [2] Algebraic equations and their applications in real-life problems.

Comments

• [1] "Great article! I was able to follow the steps and understand the solution."
• [2] "I have a question about the quadratic formula. Can you explain it in more detail?"
• [3] "I think this problem is a great example of how algebraic equations can be used to solve real-life problems."
  The Total Area of Two Square Windows: A Mathematical Puzzle - Q&A ==================================================================

Introduction

In our previous article, we solved a mathematical problem involving two square windows. The total area of the two windows is given as $1,025 \, \text{in}^2$, and we are asked to find the length of the sides of the smaller window. The problem states that each side of the larger window is 5 inches longer than the sides of the smaller window. We used algebraic equations to solve this problem and find the length of the sides of the smaller window.

Q&A

Q: What is the formula for the area of a square?

A: The formula for the area of a square is $A = s^2$, where $A$ is the area and $s$ is the length of the side.

Q: How do we find the length of the sides of the smaller window?

A: We set up an equation using the areas of the two windows and solve for the length of the side of the smaller window.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations. It is given by:

$$s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do we use the quadratic formula to solve the equation?

A: We plug in the values of $a$, $b$, and $c$ into the quadratic formula and simplify the expression under the square root.

Q: What are the possible solutions for the length of the sides of the smaller window?

A: We have two possible solutions for $s$:

$$s = \frac{-10 + 90}{4} = 20$$

$$s = \frac{-10 - 90}{4} = -25$$

Q: Why do we discard the solution $s = -25$?

A: We discard the solution $s = -25$ because the length of the side of a window cannot be negative.

Q: What is the final answer for the length of the sides of the smaller window?

A: The final answer for the length of the sides of the smaller window is $s = 20$ inches.

Frequently Asked Questions

Q: What is the total area of the two windows?

A: The total area of the two windows is $1,025 \, \text{in}^2$.

Q: How do we find the length of the sides of the larger window?

A: We find the length of the sides of the larger window by adding 5 inches to the length of the sides of the smaller window.

Q: Can we use the quadratic formula to solve any quadratic equation?

A: Yes, we can use the quadratic formula to solve any quadratic equation.

Q: What are some real-life applications of algebraic equations?

A: Algebraic equations have many real-life applications, including solving problems in physics, engineering, and economics.

Conclusion

In this article, we answered some frequently asked questions about the mathematical problem involving two square windows. We used algebraic equations to solve this problem and find the length of the sides of the smaller window. We also discussed the quadratic formula and its applications in real-life problems.

References

• [1] Algebraic equations, Wikipedia.
• [2] Quadratic formula, Wikipedia.

Related Articles

• [1] Solving quadratic equations using the quadratic formula.
• [2] Algebraic equations and their applications in real-life problems.

Comments

• [1] "Great article! I was able to follow the steps and understand the solution."
• [2] "I have a question about the quadratic formula. Can you explain it in more detail?"
• [3] "I think this problem is a great example of how algebraic equations can be used to solve real-life problems."