A photo measures 10cm by 8cm. if you wish to increase each picture of the side uniformly by such that the area increased by 75%, what will the new dimensions be
let x be the increase , assuming we increase only on one side.
(10+x)(8+x) = 1.75(80)
80 + 18x + x^2 = 140
x^2 + 18x - 60 = 0
let's complete the square ...
x^2 + 18x + 81 = 60+81
(x+9)^2 = 141
x+9 = ±√141
x = appr 2.87 cm or some inadmissible negative
the new dimension is 12.87 cm by 10.87 cm
check:
new area = 12.87..(10.87..) = 140
(I carried the full answer in my calculator)
which is a 75% increase
To find the new dimensions of the photo, we need to determine the scale factor by which the sides should be increased.
1. Start by calculating the original area of the photo.
Original Area = Length * Width = 10 cm * 8 cm = 80 cm².
2. Next, determine the factor by which the area should be increased.
Increased Area = Original Area + 75% of Original Area
Increased Area = 80 cm² + (75/100) * 80 cm²
Increased Area = 80 cm² + 60 cm² = 140 cm².
3. Now, find the scale factor by comparing the increased area with the original area.
Scale Factor = √(Increased Area / Original Area)
Scale Factor = √(140 cm² / 80 cm²)
Scale Factor = √1.75 ≈ 1.32.
4. Finally, multiply each side of the original photo by the scale factor to get the new dimensions.
New Length = Original Length * Scale Factor = 10 cm * 1.32 ≈ 13.2 cm.
New Width = Original Width * Scale Factor = 8 cm * 1.32 ≈ 10.56 cm.
Therefore, the new dimensions of the photo will be approximately 13.2 cm by 10.56 cm.