So I'm confused about this question:
Jill is designing a picture frame for an art project. She plans to have a square piece of glass in the center and surround it with a decorated ceramic frame which will also be a square. The dimension of the glass X, the dimension of the frame is X^2 - 3. Jill determines that she needs 27 square inches of material for the frame.
What are the dimensions of the glass piece?
What are the dimensions of the frame?
So the equation I got was X^4 - 7X^2 + 9 = 27
How would I at least start out those two problems? I'm not really looking for a direct answer, but I'm looking for the steps to solve it.
Thanks in advance!
Jill is designing a picture frame for an art project. She plans to have a square piece of glass in the center and surround it with a decorated ceramic frame which will also be a square. The dimension of the glass X, the dimension of the frame is X^2 - 3. Jill determines that she needs 27 square inches of material for the frame.
What are the dimensions of the glass piece?
What are the dimensions of the frame?
So the equation I got was X^4 - 7X^2 + 9 = 27
How would I at least start out those two problems? I'm not really looking for a direct answer, but I'm looking for the steps to solve it.
Thanks in advance!
-
> How would I at least start out those two problems?
It looks to me like you already did. Now just follow through with it.
If the length of each side of the frame's inner dimension (i.e. the glass) is x, and the length of each side of the frame's outer dimension is x^2 - 3, then the area of the frame is:
A = (outer dimension)^2 - (inner dimension)^2
27 = (x^2 - 3)^2 - (x)^2
27 = x^4 - 6x^2 + 9 - x^2
27 = x^4 - 7x^2 + 9 <----- This is where you got to already.
0 = x^4 - 7x^2 - 18
0 = (x^2 - 9)(x^2 + 2)
Therefore, either x^2 = 9 or x^2 = -2... and you know it can't be the latter. Thus, x^2 = 9 and therefore x = 3. The glass is 3" on each side.
Each (outer) side of the frame is x^2 - 3 = 9 - 3 = 6"
==========
A quick check of our work.
The entire frame (including picture) is 6^2 = 36.
The glass/picture portion is 3^2 = 9.
The area of the frame is 36 - 9 = 27.
Bingo!
It looks to me like you already did. Now just follow through with it.
If the length of each side of the frame's inner dimension (i.e. the glass) is x, and the length of each side of the frame's outer dimension is x^2 - 3, then the area of the frame is:
A = (outer dimension)^2 - (inner dimension)^2
27 = (x^2 - 3)^2 - (x)^2
27 = x^4 - 6x^2 + 9 - x^2
27 = x^4 - 7x^2 + 9 <----- This is where you got to already.
0 = x^4 - 7x^2 - 18
0 = (x^2 - 9)(x^2 + 2)
Therefore, either x^2 = 9 or x^2 = -2... and you know it can't be the latter. Thus, x^2 = 9 and therefore x = 3. The glass is 3" on each side.
Each (outer) side of the frame is x^2 - 3 = 9 - 3 = 6"
==========
A quick check of our work.
The entire frame (including picture) is 6^2 = 36.
The glass/picture portion is 3^2 = 9.
The area of the frame is 36 - 9 = 27.
Bingo!