A rectangular canvas picture measures 10 inches by 8 inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to 120 square inches. Find the uniform width of the frame.

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mauricioalessandrell mauricioalessandrell · Answer 1 · 2019-10-27T11:22:25+01:00

Answer:

The uniform width of the frame is 1 inch

Step-by-step explanation:

Hi there!

Please see the attached figure for a graphical description of the problem.

The area of a rectangle is calculated multiplying the length times the width.

In this case, the length of the rectangle (canvas + frame) is 10 inches + 2x (see figure), while the widht is 8 inches + 2x. Then:

area of the rectangle = (8 + 2x) · (10 + 2x) = 120

Now we have our equation. If we solve it for x, we will obtain the uniform width of the frame:

120 = (8 + 2x) · (10 + 2x)

Apply distributive property:

120 = 80 + 16x + 20x + 4x²

Subtract 120 to both sides of the equation:

0 = -120 + 80 + 16x + 20x + 4x²

0 = -40 + 36x + 4x²

Solve the quadratic equation using the quadratic formula:

a = 4

b = 36

c = -40

x = [-b ± √(b² - 4ac)]/2a

x = 1

(the second solution is negative and therefore discarded).

Then, the uniform width of the frame is 1 inch.