Question

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. Part A: Which equation represents the combined area of the photo and the mat? Part B: What is the factored form of the equation that represents the combined area of the photo and the mat? Select two answers: one for Part A and one for Part B.

Answer 1

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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