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. 13.5 In.B. 20 In.C. 25 In.D. 31.6 In.
Problem Description
We are given that the total area of two square windows is $1,025 \text{ in}^2$. Additionally, we know that each side of the larger window is 5 inches longer than the sides of the smaller window. Our goal is to find the length of the sides of the smaller window.
Step 1: Understand the relationship between the areas of the two windows
Let's assume that the side length of the smaller window is x inches. Since each side of the larger window is 5 inches longer than the sides of the smaller window, the side length of the larger window is (x + 5) inches.
Step 2: Calculate the areas of the two windows
The area of a square is given by the formula A = s^2, where s is the side length of the square. Therefore, the area of the smaller window is x^2 square inches, and the area of the larger window is (x + 5)^2 square inches.
Step 3: Set up an equation to represent the total area of the two windows
We are given that the total area of the two windows is $1,025 \text{ in}^2$. Therefore, we can set up the following equation:
x^2 + (x + 5)^2 = 1025
Step 4: Expand and simplify the equation
Expanding the equation, we get:
x^2 + x^2 + 10x + 25 = 1025
Combine like terms:
2x^2 + 10x + 25 = 1025
Subtract 1025 from both sides:
2x^2 + 10x - 1000 = 0
Step 5: Solve the quadratic equation
We can solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = 10, and c = -1000. Plugging these values into the formula, we get:
x = (-(10) ± √((10)^2 - 4(2)(-1000))) / 2(2)
x = (-10 ± √(100 + 8000)) / 4
x = (-10 ± √8100) / 4
x = (-10 ± 90) / 4
Step 6: Find the possible values of x
We have two possible values for x:
x = (-10 + 90) / 4 = 80 / 4 = 20
x = (-10 - 90) / 4 = -100 / 4 = -25
Since the side length of a window cannot be negative, we discard the negative solution.
Step 7: Check the solution
We have found that the side length of the smaller window is 20 inches. To check this solution, we can calculate the area of the smaller window and the larger window:
Area of smaller window = x^2 = 20^2 = 400 square inches
Area of larger window = (x + 5)^2 = (20 + 5)^2 = 25^2 = 625 square inches
Total area = 400 + 625 = 1025 square inches
This matches the given total area, so our solution is correct.
The final answer is:
Frequently Asked Questions
Q: What is the problem asking for?
A: The problem is asking for the length of the sides of the smaller window, given that the total area of two square 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.
Q: How do we start solving the problem?
A: We start by assuming that the side length of the smaller window is x inches. Since each side of the larger window is 5 inches longer than the sides of the smaller window, the side length of the larger window is (x + 5) inches.
Q: What is the relationship between the areas of the two windows?
A: The area of a square is given by the formula A = s^2, where s is the side length of the square. Therefore, the area of the smaller window is x^2 square inches, and the area of the larger window is (x + 5)^2 square inches.
Q: How do we set up an equation to represent the total area of the two windows?
A: We are given that the total area of the two windows is $1,025 \text{ in}^2$. Therefore, we can set up the following equation:
x^2 + (x + 5)^2 = 1025
Q: How do we solve the quadratic equation?
A: We can solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = 10, and c = -1000. Plugging these values into the formula, we get:
x = (-(10) ± √((10)^2 - 4(2)(-1000))) / 2(2)
x = (-10 ± √(100 + 8000)) / 4
x = (-10 ± √8100) / 4
x = (-10 ± 90) / 4
Q: What are the possible values of x?
A: We have two possible values for x:
x = (-10 + 90) / 4 = 80 / 4 = 20
x = (-10 - 90) / 4 = -100 / 4 = -25
Since the side length of a window cannot be negative, we discard the negative solution.
Q: How do we check the solution?
A: We have found that the side length of the smaller window is 20 inches. To check this solution, we can calculate the area of the smaller window and the larger window:
Area of smaller window = x^2 = 20^2 = 400 square inches
Area of larger window = (x + 5)^2 = (20 + 5)^2 = 25^2 = 625 square inches
Total area = 400 + 625 = 1025 square inches
This matches the given total area, so our solution is correct.
Additional Tips and Tricks
- When solving quadratic equations, make sure to check for any negative solutions, as they may not be valid in certain contexts.
- When checking solutions, make sure to calculate the area of both windows and add them together to ensure that the total area matches the given value.
- When working with square roots, make sure to simplify the expression and check for any negative solutions.
Conclusion
In this article, we have solved the problem of finding the length of the sides of the smaller window, given that the total area of two square 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. We have used the quadratic formula to solve the equation and checked the solution to ensure that it is correct.