More Word Problems Using Quadratic Equations - Example 1

Summary

In this video, the instructor solves a quadratic word problem involving the dimensions of a rectangle. Given that the area of the rectangle is 114.4 square meters and the length is 14 meters longer than the width, the instructor demonstrates how to set up and solve the quadratic equation using the quadratic formula. The process involves finding the width and then calculating the length, with the final dimensions being approximately 5.78 meters for the width and 19.78 meters for the length.

Takeaways

• 🟥 The problem involves finding the length and width of a rectangle with an area of 114.4 square meters.
• 📏 The length of the rectangle is 14 meters longer than its width.
• ✏️ The width is denoted as 'W', and the length is expressed as 'W + 14'.
• 📐 The area formula for a rectangle (length × width) is used: (W + 14) × W = 114.4.
• 🧮 Distributing W in the equation results in a quadratic: W^2 + 14W = 114.4.
• ➖ To form the standard quadratic equation, 114.4 is subtracted from both sides, leading to W^2 + 14W - 114.4 = 0.
• 🧑‍🏫 The quadratic formula is applied: W = (-B ± √(B^2 - 4AC)) / 2A.
• 🔢 Plugging values into the formula gives: W = (-14 ± √(196 + 457.6)) / 2.
• 🧩 Solving the equation yields an approximate width of 5.783 meters.
• 📏 The corresponding length is found to be approximately 19.783 meters (W + 14).

Q & A

What is the area of the rectangle mentioned in the problem?

The area of the rectangle is 114.4 square meters.

How is the length of the rectangle related to its width?

The length of the rectangle is 14 meters longer than its width.

What equation is used to represent the relationship between the area, length, and width of the rectangle?

The equation used is the area = length × width, which becomes 114.4 = (W + 14) × W.

How is the quadratic equation formed from the area equation?

The quadratic equation is formed by expanding and rearranging the area equation to W^2 + 14W - 114.4 = 0.

What method is used to solve the quadratic equation?

The quadratic formula is used to solve the equation: W = (-B ± √(B² - 4AC)) / 2A.

What are the values for A, B, and C in the quadratic equation?

In the equation W^2 + 14W - 114.4 = 0, A = 1, B = 14, and C = -114.4.

What is the result after calculating the discriminant (B² - 4AC)?

The discriminant is calculated as 653.6.

Which solution from the quadratic formula is used and why?

The positive solution is used because the width cannot be a negative value. The solution used is approximately 5.783 meters.

What is the width of the rectangle based on the solution?

The width of the rectangle is approximately 5.783 meters.

What is the length of the rectangle after solving the problem?

The length of the rectangle is approximately 19.783 meters, found by adding 14 meters to the width.