Question

A rectangular picture frame is 6 inches wide and 10 inches tall. You want to make

the area 7 times as large by increasing the length and width by the same amount.
Find the number of inchey by which each dimension must be increased.

Answer 1

Explanation:

let length and width be increased by x>0

(10+x)(5+x)=6×10×5

50+10 x+5x+x²=300

x²+15x-250=0

x²+25 x-10x-250=0

x(x+25)-10(x+25)=0

(x+25)(x-10)=0

x=-25(rejected),10

so x=10

so both the dimensions should be increase by 10 inches.

Answer 2

Let x be the number of inches by which each dimension must be increased. Then the new width of the picture frame is 6 + x inches, and the new height is 10 + x inches. The new area of the picture frame is 7 times the original area, which is:

7(6 in × 10 in) = 420 in²

The new area can also be expressed as the product of the new length and width:

(6 in + x)(10 in + x)

Expanding this expression, we get:

60 in² + 16x + x² = 420 in²

Subtracting 420 in² from both sides and simplifying, we get:

x² + 16x - 360 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-16 ± sqrt(16² + 4(1)(360))) / 2(1)

x = (-16 ± sqrt(256 + 1440)) / 2

x = (-16 ± sqrt(1696)) / 2

x = (-16 ± 41.19) / 2

x = -28.59 / 2 or x = 25.59 / 2

Since we are looking for a positive increase in dimension, we discard the negative solution and obtain:

x = 25.59 / 2 ≈ 12.795

Therefore, each dimension must be increased by approximately 12.795 inches to make the area 7 times as large.