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A rectangular picture frame is 5 inches wide and 10 inches tall. You want to make the area 6 times as large by increasing the length and width by the same amount. Find the number of inches by which each dimension must be increased.

A rectangular picture frame is 55 inches wide and 1010 inches tall. You want to make the area 66 times as large by increasing the length and width by the same amount. Find the number of inches by which each dimension must be increased.

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Full solution

Q. A rectangular picture frame is 55 inches wide and 1010 inches tall. You want to make the area 66 times as large by increasing the length and width by the same amount. Find the number of inches by which each dimension must be increased.
  1. Calculate original area: Original area of the frame is 55 inches ×\times 1010 inches.\newlineCalculate the original area: 5×10=505 \times 10 = 50 square inches.
  2. Calculate new area: The new area should be 66 times the original area.\newlineCalculate the new area: 6×50=3006 \times 50 = 300 square inches.
  3. Find new dimensions: Let xx be the number of inches to increase both the length and width.\newlineThe new dimensions will be (5+x)(5 + x) inches by (10+x)(10 + x) inches.
  4. Set up new area equation: The new area is (5+x)×(10+x)(5 + x) \times (10 + x).\newlineSet up the equation for the new area: (5+x)(10+x)=300(5 + x)(10 + x) = 300.
  5. Expand and simplify equation: Expand the equation: 50+5x+10x+x2=30050 + 5x + 10x + x^2 = 300. Simplify the equation: x2+15x+50=300x^2 + 15x + 50 = 300.
  6. Set equation to zero: Subtract 300300 from both sides to set the equation to zero: x2+15x+50−300=0x^2 + 15x + 50 - 300 = 0.\newlineSimplify the equation: x2+15x−250=0x^2 + 15x - 250 = 0.
  7. Factor quadratic equation: Factor the quadratic equation: (x+25)(x−10)=0(x + 25)(x - 10) = 0.

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