A photograph measures 20cm by 16cm.
A strip of constant width is to be cut off the top and one side of the photo, so the area is reduced to 60% of the area of the original photo. Find the width of the cut.
Thank you so much!
(20-2w)(16-2w) = .60*20*16
To find the width of the cut, we can start by calculating the original area of the photograph.
Original area = Length × Width
Given that the length is 20cm and the width is 16cm, the original area is:
Original area = 20cm × 16cm = 320cm²
Next, we need to find the reduced area, which is 60% of the original area:
Reduced area = 60% of Original area
Reduced area = 0.60 × Original area
Reduced area = 0.60 × 320cm² = 192cm²
Now, we can set up an equation to find the width of the cut.
Let's assume the width of the cut is x cm.
The new width of the photograph after the cut will be (16 - x) cm, and the new length will be (20 - x) cm.
The new area of the photograph after the cut is:
New area = (16 - x) cm × (20 - x) cm
Since we know the reduced area is 192cm², we can set up the following equation:
(16 - x) cm × (20 - x) cm = 192cm²
Simplifying the equation, we have:
(320 - 36x + x²) cm² = 192cm²
Now, we can solve for x by rearranging the equation and solving the quadratic equation:
x² - 36x + 128 = 0
Using the quadratic formula x = (-b ± sqrt(b² - 4ac)) / (2a), where a = 1, b = -36, and c = 128, we can find the values of x.
x = (-(-36) ± sqrt((-36)² - 4(1)(128))) / (2(1))
Simplifying further, we have:
x = (36 ± sqrt(1296 - 512)) / 2
x = (36 ± sqrt(784)) / 2
x = (36 ± 28) / 2
The two possible values for x are:
x₁ = (36 + 28) / 2 = 32cm
x₂ = (36 - 28) / 2 = 4cm
Since the width of the cut cannot be negative, the width of the cut is 4cm.
Therefore, the width of the cut is 4cm.
To solve this problem, we need to find the width of the strip that needs to be cut off the top and side of the photograph. Let's break down the problem into steps:
Step 1: Find the original area of the photograph.
The area of a rectangle is calculated by multiplying its length and width. In this case, the length of the photograph is 20cm and the width is 16cm. So, the original area of the photograph is 20cm * 16cm = 320cm².
Step 2: Find the desired area of the reduced photograph.
The problem states that the area of the photograph needs to be reduced to 60% of the original area. To find this desired area, we multiply the original area by 0.6:
320cm² * 0.6 = 192cm².
Step 3: Set up an equation with the unknown width of the cut.
Let's assume the width of the cut is 'x' cm. If we cut off a strip from the top and one side, the new length of the photograph will be reduced by 'x' cm, and the new width will be reduced by 'x' cm as well. So, the new dimensions of the photograph will be (20cm - x) by (16cm - x). And the area of the new photograph will be (20cm - x) * (16cm - x).
Step 4: Solve the equation.
We want the area of the new photograph to be equal to the desired area of 192cm². So, we can set up the equation:
(20cm - x) * (16cm - x) = 192cm².
Step 5: Solve for 'x'.
Expand the equation:
320cm² - 20cmx - 16cmx + x² = 192cm².
Rearrange the equation and combine like terms:
x² - 36cmx + 320cm² - 192cm² = 0.
x² - 36cmx + 128cm² = 0.
Now, we can solve this quadratic equation to find the values of 'x'. You can use factoring, completing the square, or the quadratic formula to solve this equation.