SOLUTION: A framed painting has overall dimensions of 32"x29". the area of the painting itself is 550sq". Find the width of the fame. Please show the work. Thank you!

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: A framed painting has overall dimensions of 32"x29". the area of the painting itself is 550sq". Find the width of the fame. Please show the work. Thank you!      Log On


   

Question 1074358: A framed painting has overall dimensions of 32"x29". the area of the painting itself is 550sq". Find the width of the fame. Please show the work. Thank you!
Found 2 solutions by josmiceli, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
A key to this question is that they are assuming the
width of the frame is uniform all around
Let the width of the frame = +x+
------------------------------------------------
I will subtract the width of the frame from BOTH sides
of the overall width AND the overall height to get the
dimensions of the painting itself
--------------------------------------------------
+%28+32+-+2x+%29%2A%28+29+-+2x+%29+=+550+
+928++-+58x+-+64x+%2B+4x%5E2+=+550+
+4x%5E2+-+122x+%2B+378+=+0+
+2x%5E2+-+61x+%2B+189+=+0+
Use the quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+2+
+b+=+-61+
+c+=+189+
x+=+%28-%28-61%29+%2B-+sqrt%28+%28-61%29%5E2-4%2A2%2A189+%29%29%2F%282%2A2%29+
x+=+%28+61+%2B-+sqrt%28+3721+-+1512+%29%29%2F4+
x+=+%28+61+%2B-+sqrt%28+2209+%29%29%2F4+
+x+=+%28+61+-+47+%29%2F4+ ( the positive square root is too big )
+x+=+14%2F4+
+x+=+3.5+
The width of the frame is 3.5 inches
---------------------------------
check:
+%28+32+-+2x+%29%2A%28+29+-+2x+%29+=+550+
+%28+32+-+2%2A3.5+%29%2A%28+29+-+2%2A3.5+%29+=+550+
+%28+32+-+7+%29%2A%28+29+-+7+%29+=+550+
+25%2A22+=+550+
+550+=+550+
OK

Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.
Assuming the width of the frame uniform, let x be the frame width.


Then the dimensions of the painting itself are 32-2x and 29-2x inches.

The area equation for the painting is

(32-2x)*(29-2x) = 550.

Simplify and solve for x.


For many other similar solved problems see the lesson
    - Problems on the area and the dimensions of a rectangle surrounded by a strip
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Dimensions and the area of rectangles and circles and their elements".