Picture Frame Problem Solution

Summary

The video explains a mathematical problem involving an 8x10 inch photograph reduced in size with a uniform border around it. The goal is to determine the width of the border, given that the area of the reduced photograph is 63 square inches. The instructor walks through the process of visualizing the problem, setting up an equation based on the dimensions, and solving the quadratic equation using the quadratic formula. After solving, the correct width of the border is determined to be 1/2 inch, as the other possible solution is not feasible.

Takeaways

• 📏 The problem involves reducing an 8x10 inch photograph with a uniform-width border around it.
• 🖼️ The area of the reduced photograph is 63 square inches.
• ✏️ To solve the problem, first draw the photograph and its border for better visualization.
• 🔍 Let the width of the border be represented by X.
• 🔢 The length and width of the reduced photograph are calculated as (10 - 2X) and (8 - 2X), respectively.
• 🧮 The area equation becomes: (10 - 2X) * (8 - 2X) = 63 square inches.
• 📐 Expanding the equation gives: 4X² - 36X + 80 = 63.
• 📉 Simplifying the equation leads to: 4X² - 36X + 17 = 0, a standard quadratic form.
• 🧑‍🏫 The quadratic formula is used to solve for X, with two possible solutions: X = 8.5 inches or X = 0.5 inches.
• ✅ After evaluating both answers, only X = 0.5 inches makes sense for the border width.

Q & A

What is the size of the original photograph?

The original photograph is 8 inches by 10 inches.

What is the area of the reduced photograph?

The area of the reduced photograph is 63 square inches.

What variable is used to represent the width of the border?

The variable 'X' is used to represent the width of the border.

How is the length of the reduced photograph expressed in terms of X?

The length of the reduced photograph is expressed as 10 - 2X.

How is the width of the reduced photograph expressed in terms of X?

The width of the reduced photograph is expressed as 8 - 2X.

What equation is used to represent the area of the reduced photograph?

The equation for the area is (10 - 2X) * (8 - 2X) = 63.

What is the expanded form of the area equation?

The expanded form of the equation is 4X² - 36X + 80 = 63.

How is the quadratic equation formed from the area equation?

The quadratic equation is formed by subtracting 63 from both sides: 4X² - 36X + 17 = 0.

What are the values of X obtained using the quadratic formula?

The values of X are 8.5 inches and 0.5 inches.

Which value of X is valid for the width of the border and why?

The valid value of X is 0.5 inches because a border width of 8.5 inches would be larger than the original photograph, which does not make sense.