Picture Frame Problem Solution
Summary
TLDRThe 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
🖼️ 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.
📏 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.
Mindmap
Keywords
💡Photograph
💡Reduced in size
💡Centered
💡Uniform width
💡Area
💡Length
💡Width
💡Equation
💡Quadratic formula
💡Solve
💡Contextualize
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
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