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.

Outlines

00:00

🖼️ Understanding the Problem: Reducing the Photograph Size

In this paragraph, the speaker introduces the problem, which involves an 8x10-inch photograph being reduced in size while keeping it centered with a uniform border around it. The goal is to find the width of the border given that the area of the reduced photograph is 63 square inches. The speaker emphasizes the importance of visualizing the problem by drawing the photograph, labeling its dimensions, and assigning variables. The width of the border is represented by 'X,' and an equation is set up to find the reduced photograph’s length and width.

05:24

📏 Setting Up the Equation: Length, Width, and Area

Here, the speaker constructs the mathematical equation based on the given conditions. They calculate the reduced photograph’s length as 10 - 2X and the width as 8 - 2X, using these expressions to represent the reduced photograph’s area as (10 - 2X) * (8 - 2X). Since the area is given as 63 square inches, this information is used to form a quadratic equation. The speaker then begins solving the quadratic equation by expanding it, combining like terms, and preparing it for solution.

🧮 Solving the Quadratic Equation

The speaker continues by solving the quadratic equation, identifying the coefficients for the quadratic formula: a = 4, b = -36, and c = 17. After applying the quadratic formula, two possible solutions for X are found: 8.5 inches and 0.5 inches. However, since a border width of 8.5 inches is not realistic given the dimensions of the photograph, the only feasible solution is a border width of 0.5 inches.

✅ Final Answer: Border Width

In the final paragraph, the speaker concludes that the correct width of the border is 0.5 inches. They validate this answer by explaining why the alternative solution (8.5 inches) does not make sense, as it would be larger than the photograph's width. The speaker closes by confidently stating that the border is 0.5 inches wide.

Keywords

💡Photograph

A photograph is an image created by focusing light onto a light-sensitive surface, usually photographic film or a digital sensor. In the video's context, it refers to an 8 by 10 inch photograph that is being resized. The photograph serves as the central object in the problem-solving scenario presented in the video.

💡Reduced in size

Reducing in size refers to the process of making something smaller in dimensions. In the video, the photograph is reduced in size to fit onto a page with a uniform border around it. This concept is crucial to understanding the mathematical problem that the video aims to solve.

💡Centered

To center something means to place it in the middle of an area. In the video, the reduced photograph is centered on the page, which is an important detail for visualizing the problem and calculating the dimensions of the border.

💡Uniform width

Uniform width implies that all sides of a border have the same measurement. The video discusses a border of uniform width around the photograph, which is a key piece of information for setting up the equation to solve for the border's width.

💡Area

Area is a measure of the space enclosed within a shape, typically expressed in square units. The video mentions the area of the reduced photograph being 63 square inches, which is the basis for the mathematical equation to find the border's width.

💡Length

Length refers to the measurement of distance or extent of something from end to end. In the video, the length of the reduced photograph is calculated by subtracting twice the border width from the total length of the page (10 inches).

💡Width

Width is a measure of how much space something occupies from side to side. The video calculates the width of the reduced photograph by subtracting twice the border width from the total width of the page (8 inches).

💡Equation

An equation is a mathematical statement that asserts the equality of two expressions. The video sets up an equation to represent the relationship between the dimensions of the photograph and the area after reduction, which is essential for finding the border's width.

💡Quadratic formula

The quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0. In the video, the quadratic formula is applied to find the width of the border, which is represented by the variable x in the equation.

💡Solve

To solve, in a mathematical context, means to find the answer or solution to a problem. The video demonstrates the process of solving for the border width by using the quadratic formula, which is the crux of the mathematical problem presented.

💡Contextualize

Contextualize means to relate or adapt something to a specific situation or context. In the video, the problem is contextualized by visualizing the photograph and the border, which helps in understanding and solving the mathematical equation.

Highlights

The problem involves reducing the size of an 8 by 10 inch photograph to fit on a page with a uniform border.

The area of the reduced photograph is given as 63 square inches.

The border width around the photograph is uniform and needs to be determined.

The first step is to visualize the problem by drawing a diagram.

The original photograph dimensions are 8 inches by 10 inches.

The reduced photograph is centered on the page with a uniform border.

The width of the border is represented by the variable X.

The area of the reduced photograph is calculated as the product of its length and width.

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

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

The equation for the area of the reduced photograph is set up as (10 - 2X)(8 - 2X) = 63.

The equation is expanded to 4X^2 - 36X + 80 = 63.

The equation is rearranged into standard quadratic form 4X^2 - 36X + 17 = 0.

The quadratic formula is used to solve for X.

The parameters for the quadratic formula are identified as a=4, b=-36, and c=17.

Two potential solutions for X are calculated: 8.5 inches and 0.5 inches.

The solution of 8.5 inches is discarded as it is not practical for the given dimensions.

The final answer is that the border width is 0.5 inches.

Transcripts

so at first glance this problem seems

quite difficult but really the hardest

part about this problem is actually

picturing what it is you're being asked

so let's look at this so it says an 8 by

10 photograph is reduced in size oops an

8 by 10 inch photograph is reduced in

size so that it is centered on the page

with the border of uniform width around

it if the area of the reduced photograph

is 63 square inches determine the width

of the border okay so the first thing

you want to do is you want to draw this

okay and that's really the easiest way

thing to do so let's make a nice let's

make a picture here so here's the

original photo okay so the original

photo is 8 inches here and 10 inches

here and it says it's reduced in size so

that it's centered on the page so here's

the reduced version of the photo with a

unit border of uniform width so this is

the width this is the width and this is

the width and this is the width okay

so let's let the width of the border be

X okay so we have X X X X all right we

want the area of the reduced photograph

to be 63 square inches so this area here

is 63 inches squared all right so let's

put together our equation first of all

let's look at we know area is length

times width okay so what is this length

and this width all right so this length

here let's do this so this life here is

that Plus this Plus this equals 10

inches but those are both X so we know

that the length equals 10 minus X minus

X which is 10 minus 2x and we can do the

same thing with the width okay so it's

this Plus this plus the middle part is 8

and each one of those blue bits is X so

the width

equals 10 - sorry not 10 8 minus X minus

X which is 8 minus 2x all right so the

area of the reduced photograph is length

times width which is 10 minus 2x times 8

minus 2x and we also know the area

equals 63 inches squared all right so

now we have an equation let's put that

all together all right so we have our

equation and now we need to expand

collect our terms and solve for X so

first things let's expand the left side

so we get 80 minus 20x minus 16x plus 4x

squared equals 63 let's combine all our

terms so we get 4x squared minus 36 X

plus 80 equals 63 and we want to solve

this but in order to solve that we need

to have it in the form of ax squared

plus BX plus C equals 0 so we need to

move this over to the other side so we

get 4x squared minus 36

whoops 36 X plus 80 minus 63 equals 0 or

4x squared minus 36 x plus 17 equals 0

so now it's in standard form and we can

solve for X in three different ways we

can solve by completing the square we

consult by factoring or we can solve

with the quadratic formula so that's

what I'm going to do solve using the

quadratic formula so the first thing I

want to do is identify my parameter so a

equals 4 B equals negative 36 and C

equals 17 so now let's solve for x

you

all right so there's two values of X X

equals 36 plus 32 over 8 or x equals 36

minus 32 over 8 so this is 68 over 8

which is 8 and 1/2 inches and this one

would be 4 over 8 which is 1/2 inch so

you need so we have two answers so X

either equals 8 and 1/2 inches or what

and half of an inch so now we're you

slide on your final answers when you

have to go back to yourself and say well

which answer makes the most sense all

right so let's say we took eight and a

half inches so we go back to your

picture and if we took away I mean that

doesn't make sense because the whole

picture here is only 8 inches so we

can't have it at 8 and 1/2 inches so

this one clearly does not make sense

this value however does all right so we

have one value so then you need a final

statement the border is 1/2 inch wide

okay so there you go