Leonard needs to design a mat with an 8-×10-inch opening for a photo that he is going to frame. He wants the mat to have the same width on all four sides, as shown in the following figure.
A small 8-inch by 10-inch rectangle inside a larger rectangle. The distance between the two rectangles is labeled w.
The combined area of the photo and mat needs to be 224 square inches. The equation that represents the combined area is (2w+8)(2w+10)=224, where w represents the width of the mat around the photo.
What should the width of the mat be?
Answers
Given : (2w+8)(2w+10)=224
To Find : w
Solution:
(2w+8)(2w+10)=224
=> 2(w + 4)*2(w + 5) = 224
=> 4(w + 4)(w + 5) = 224
=> (w + 4)( w + 5) = 56
=> w² + 9w + 20 = 56
=> w² + 9w - 36 = 0
=> w² + 12w - 3w - 36 = 0
=> w(w + 12) - 3(w + 12) = 0
=> (w - 3)(w + 12) = 0
=> w = 3 or - 12
-12 not possible as width can not be negative
Hence w= 3 ,
Width of the mat around the photo = 3
width of the mat = 3
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Given :- Leonard needs to design a mat with an 8×10 opening for a photo that he is going to frame. He wants the mat to have the same width on all four sides, as shown in the following figure. The combined area of the photo and the mat needs to be 224 square inches.
To Find :-
a) Which equation represents the combined area of the photo and the mat ?
b) What is the factored form of the equation that represents the combined area of the photo and the mat ?
c) find the width of the frame ?
Solution :-
Let us assume that, the width of the frame is x inch .
then,
→ Length of Photo with frame = Original length of photo + 2 * (width of the frame) = 10 + 2 * x = (10 + 2x) inch.
similarly,
→ Breadth of Photo with frame = Original Breadth + 2 * (width of the frame) = 8 + 2 * x = (8 + 2x) inch .
therefore,
→ Area of the Photo with Frame = Length * Breadth = (10 + 2x) * (8 + 2x) = 80 + 20x + 16x + 4x² = (4x² + 36x + 80) .
now, we have given that,
→ The combined area of the photo and the mat = 224 inch²
→ 4x² + 36x + 80 = 224
→ 4(x² + 9x + 20) = 224
→ x² + 9x + 20 = 56
→ x² + 9x + 20 - 56 = 0
→ x² + 9x - 36 = 0
→ x² + 12x - 3x - 36 = 0
→ x(x + 12) - 3(x + 12) = 0
→ (x + 12)(x - 3) = 0
→ x = (-12) or 3.
since width can't be in negative . then, value of x will be 3.
hence,
- Answer a) => (10 + 2x) * (8 + 2x) = 80 + 20x + 16x + 4x² = (4x² + 36x + 80) which is equal to 224 and if we solve this, the quadratic equation we get is = x² + 9x - 36 = 0 . (for complete solution , look up.)
- Answer b) => (10 + 2x) * (8 + 2x) = 224 or, (x + 12)(x - 3) = 0
- Answer c) => The width of the frame is 3 inch.
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