Tip:
Highlight text to annotate it
X
(male narrator) In this video,
we will be looking at the area of picture frames.
To help us visualize the frame,
we will always want to draw a picture.
We want to remember as we set up the problem,
the frame is on the top and bottom of the picture.
It is also on the left and right sides
and must be considered on both sides, not just one.
So in this example, which describes a picture...
which measures 10 inches by 7 inches,
is placed in a frame of uniform width.
We're looking for the width of the frame,
which we don't know.
Let's call it x.
This x is on both sides of the picture,
so when I want to describe the top of the picture frame here,
we have a 7, an x, and an x, or a total of 7 plus 2x.
Similarly, the frame is on the top and bottom,
so the height of the frame becomes 10 plus 2x.
We're told that the total area
of the frame and picture together is 208.
If we multiply width times length,
this will give us the area:
7 plus 2x, times 10, plus 2x, is equal to the area of 208.
We can start solving this equation
by multiplying it using FOIL
to get 70, plus 14x, plus 20x, plus 4x squared, equals 208.
Combining like terms and putting things in order gives us:
4x squared, plus 34x, plus 70, equals 208.
In order to solve, we want the equation to equal 0,
so we will subtract 208 from both sides.
This gives us 4x squared, plus 34x, minus 138, equals 0.
We can now start factoring in order to solve it
by factoring out the GCF of 2 to get 2x squared, plus 17x,
plus 16...or minus 69... equals 0.
We can continue factoring... this expression
to get 2x, plus 23, times x, minus 3, equals 0.
If you could not find those two factors,
we could've used the quadratic formula on this trinomial,
using a as 2, b as 17, and c as -69.
Both will give us the same final result.
Once it's factored,
we set each factor equal to 0 that has a variable in it.
Now, we can solve the remaining equation
by subtracting 23 to get 2x equals -23,
and dividing by 2 to get x equals -23/2;
or solving the other equation
by adding 3 to get x is equal to 3.
Remember, x represents the width of the frame.
It would not have a negative width to it,
so we can throw the negative number out.
The only answer left for the width of the frame
is x equals 3,
telling us our frame has a width of 3 inches.
In Part 2 of this video, we'll take a look at another example
where we have a frame
and are asked to find the width of the frame.
As we set these problems up, it is important to remember
that the frame is on the left and right sides,
giving us 2x in the top and bottom sides.