A square sheet of paper has area 15cm^2. The front side is white and the back side is black. A corner of the sheet is lifted and placed so that the crease is at a 45degree angle. If the fold is such that the visible black area is equal to the visible white area, how many centimeters long is the crease? Express your answer in simplest radical form.
@ Not sure, I think you forgot to divide by 2 for the area of the triangle.
The correct answer is 2\sqrt{5}
Well, this seems like quite a crafty problem! Let's unfold it, shall we?
First, we know that the square sheet of paper has an area of 15 cm^2. Since it's a square, that means each side is √15 cm long.
Now, let's consider the white and black areas after the fold. The visible black area must be equal to the visible white area. Since the total area is 15 cm^2, that means each side should have an area of 7.5 cm^2.
Now, let's imagine that we unfold the paper. We'll have a right triangle with one leg being the side of the square (√15 cm) and the hypotenuse being the crease. We want to find the length of the other leg, which represents the visible white area.
Using the Pythagorean theorem, we have:
(√15 cm)^2 = (√15 cm - leg length)^2 + (leg length)^2
15 cm = 15 cm - 2√15 cm * leg length + leg length^2 + leg length^2
30 cm * leg length - 2√15 cm * (leg length)^2 = 0
Dividing through by 2√15 cm, we get:
15 cm * leg length - (leg length)^2 = 0
Rearranging the equation, we have:
(leg length)^2 - 15 cm * leg length = 0
Factoring out leg length, we have:
leg length * (leg length - 15 cm) = 0
So, we have two possible solutions: leg length = 0 or leg length = 15 cm.
Since having a crease of length 0 cm doesn't seem physically possible, we can conclude that the crease must be 15 cm long.
Therefore, the length of the crease is 15 cm in simplest radical form.
I hope my folding skills didn't leave you in a twist!
Let's assume the length of each side of the square sheet of paper is "s" cm.
The area of a square is given by the formula: Area = side^2.
Given that the area of the square sheet is 15 cm^2, we can write the equation as:
s^2 = 15 (Equation 1)
Now, let's consider the visible black and white areas after folding.
When the crease is made at a 45-degree angle, it divides the square sheet into two right-angled triangles and a small square in the corner.
The area of the small square can be calculated as (s/2)^2 = (s^2)/4.
The area of one right-angled triangle will be half the area of the small square, which is (s^2)/8.
Since the visible black area is equal to the visible white area, we can set up the following equation:
Black area = White area
((s^2)/8) + (s^2)/4 = ((s^2)/8) + (s^2)/4
Multiplying each term by 8 to eliminate the denominators:
s^2 + 2s^2 = s^2 + 2s^2
3s^2 = 3s^2
Canceling out the common terms, we get:
s^2 = s^2
This equation shows that the length of the side of the square sheet (s) is unchanged after folding.
So, the length of the crease will be equal to the length of one side of the square sheet, which is "s" cm.
Now, let's find the value of "s" by using Equation 1:
s^2 = 15
Taking the square root of both sides:
s = √15
Therefore, the length of the crease is √15 cm (in simplest radical form).
To solve this problem, we need to find the length of the crease when the visible black area is equal to the visible white area.
First, let's consider the original sheet of paper:
- The area of the square sheet of paper is 15 cm^2. Since it's a square, the length of each side is √(15) cm.
Next, let's consider the area of the visible white and black regions after folding the sheet:
- When we fold the sheet along the crease, we form a right triangle. One leg of this triangle is the length of the crease, and the other two legs are the sides of the square sheet of paper.
- The white area that becomes visible after folding is the area of this right triangle, which is (1/2) * base * height. In this case, the base length is equal to the length of the crease, and the height is the length of the side of the square sheet of paper, which is √(15) cm.
- Similarly, the black area that becomes visible after folding is also the area of this right triangle, which is also (1/2) * base * height. In this case, the base length is equal to the length of the crease, and the height is the length of the side of the square sheet of paper, which is √(15) cm.
- We know that the visible black area is equal to the visible white area, so we can set up an equation: (1/2) * base * height (white) = (1/2) * base * height (black)
- Simplifying the equation, we get: (√(15) * base)/2 = (√(15) * base)/2
- The base length cancels out, and we are left with: (√(15))/2 = (√(15))/2
From this equation, we can see that the length of the crease is (√(15))/2 cm.