niheil wrote:
This is a tough one. Can anyone help me out with this:
The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture, in inches?
(A) \(9\sqrt{2}\)
(B) \(\frac{3}{2}\)
(C) \(\frac{9}{\sqrt{2}}\)
(D) \(15(1-\frac{1}{\sqrt{2}})\)
(E) \(\frac{9}{2}\)
Additional Info on the Problem
Source: Paper Test
Test Code 28
Section 5
# 15
Attachment:
rectangular_picture_frame.PNG [ 10.34 KiB | Viewed 7625 times ]
Given:
Length of the frame(outer side of the black portion) = 18 inches
Width of the frame(outer side of the black portion) = 15 inches
Total Area of the frame(black portion) and picture(orange portion) = length*width = 18*15
\(A_t=18*15\)
Let the length of the picture(orange portion) be "l", we need to find this.
Let the width of the picture(orange portion) be "w"
Area of the picture(orange portion) = l*w
\(A_p=l*w\)
Area of the frame(black portion) = Total Area of the frame(black) and picture(orange) - Area of the picture(orange)
\(A_f=A_t-A_p\)
"The frame encloses a rectangular picture that has the same area as the frame itself"\(A_p=A_f\)
\(A_p=A_t-A_p\)
\(A_t=2A_p\)
\(2A_p=18*15\)
\(A_p=\frac{18*15}{2}\)
\(l*w=\frac{18*15}{2}\) -----------------------1
"length and width of the picture(orange) have the same ratio as the length and width of the frame(black)"\(\frac{l}{w}=\frac{18}{15}\)
\(w=\frac{15}{18}*l\) --------------------2
Substituting "w" from 2 in 1:
\(l*\frac{15}{18}*l=\frac{18*15}{2}\)
\(l^2=\frac{(18)^2}{2}\)
Taking the square root on both sides:
\(l=\frac{18}{\sqrt{2}}\)
\(l=\frac{2*9}{\sqrt{2}}\)
\(l=\frac{\sqrt{2}*\sqrt{2}*9}{\sqrt{2}}\)
\(l=9\sqrt{2}\)
Ans: "A"
_________________
~fluke
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