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# Geometry Question. I would appreciate the help please.

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The diagram below represents a photograph in a frame. The photo has a perimeter of 23 inches, and the area of the whole figure is 58.5 square inches. If the frame is 1 inch thick, what is the area of the photo in square inches?

May 27, 2021

#1
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ok, lets say the length and width of the photo is l and w.

then, $$2\left(l+w\right)=23$$

and if the frame is 1 inch, $$\left(l+1+1\right)\left(w+1+1\right)=58.5$$

then we just solve the system of equations and get l = 7 and w = 4.5 so the area is 31.5

May 27, 2021
edited by mworkhard222  May 27, 2021
#2
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Thanks for your help but how did you solve the equations? I can't seem to get a whole number?

Guest May 27, 2021
#3
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Do you understand what mworkhard has presented so far?  I will assume so ....

2(L+W) = 23 (1)

(L+2)(W+2)=58.5  (2)

LW+2(W+L)+4=58.5

sub in (1)

LW+23+4=58.5

LW+27=58.5

LW=31.5

L=31.5/W

sub into (1) to find W

2(L+W) = 23 (1)

$$2(\frac{31.5}{W}+W) = 23 \\ \text{mult both sides by W}\\ 2W(\frac{31.5}{W}+W) = 23W \\ 63+2W^2=23W\\ 2w^2-23w+63=0\\ w=\frac{23\pm\sqrt{529-504}}{4}\\ w=\frac{23\pm\sqrt{25}}{4}\\ w=\frac{23\pm5}{4}\\ w=4.5\;or\; 7$$

Length = 7  width =4.5 inches

May 28, 2021
#4
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omg how did i not see this! Here is how u can solve the system:

1. Isolate $$l$$ in $$2(l+w)=23$$ and get $$l=\frac{23}{2}-w$$
2. Plug the solutions $$l=\frac{23}{2}-w$$ into $$\left(l+1+1\right)\left(w+1+1\right)=58.5$$
3. For $$\left(l+1+1\right)\left(w+1+1\right)=58.5$$ subsitute $$l$$ with $$\frac{23}{2}-w: \:\:w=\frac{9}{2},l=7$$

Jun 1, 2021