A rectangular photograph is mounted on a rectangular card. There is a border ofequal width around the photograph. The perimeter of the card is 40 centimeters longer than that of the photograph. Find the width of the border in centimeters.
Not enough data to solve. For example, since it not a square, what is relationship between length and width?
Let's assume the length of the photograph is "L" centimeters and the width is "W" centimeters.
Given that there is a border of equal width around the photograph, we can set up the following equations:
Length of the card = Length of the photograph + 2 * Width of the border
Width of the card = Width of the photograph + 2 * Width of the border
We are also given that the perimeter of the card is 40 centimeters longer than that of the photograph. The perimeter of a rectangle is given by the formula:
Perimeter = 2 * (Length + Width)
So the perimeter of the card is:
Perimeter of the card = 2 * (Length of the card + Width of the card)
And the perimeter of the photograph is:
Perimeter of the photograph = 2 * (Length of the photograph + Width of the photograph)
According to the problem, the perimeter of the card is 40 centimeters longer than the perimeter of the photograph. Therefore, we can write the equation as:
Perimeter of the card = Perimeter of the photograph + 40
Substituting the formulas for the perimeters, we get:
2 * (Length of the card + Width of the card) = 2 * (Length of the photograph + Width of the photograph) + 40
Now we can substitute the formulas for the length and width of the card and photograph:
2 * ((Length of the photograph + 2 * Width of the border) + (Width of the photograph + 2 * Width of the border)) = 2 * (Length of the photograph + Width of the photograph) + 40
Simplifying the equation:
2 * (Length of the photograph + Width of the photograph + 2 * Width of the border) = 2 * (Length of the photograph + Width of the photograph) + 40
Now we can simplify further:
2 * Length of the photograph + 2 * Width of the photograph + 4 * Width of the border = 2 * Length of the photograph + 2 * Width of the photograph + 40
We can cancel out the common terms:
4 * Width of the border = 40
Simplifying the equation:
Width of the border = 40 / 4
Therefore, the width of the border is 10 centimeters.
To find the width of the border, we need to understand the relationship between the dimensions of the photograph, the dimensions of the card, and the width of the border.
Let's assume the width of the border is x centimeters.
The photograph will have two dimensions: length and width. Let's call the length of the photograph L and the width of the photograph W.
Now, we know that the dimensions of the card will be increased by twice the width of the border, because there are two borders (top/bottom and left/right).
So, the length of the card will be L + 2x, and the width of the card will be W + 2x.
The perimeter of the photograph can be calculated by adding up all the sides, which is:
2L + 2W.
Similarly, the perimeter of the card can be calculated as:
2(L + 2x) + 2(W + 2x).
According to the problem, the perimeter of the card is 40 centimeters longer than the perimeter of the photograph:
2(L + 2x) + 2(W + 2x) = 2L + 2W + 40.
Simplifying this equation, we get:
2L + 4x + 2W + 4x = 2L + 2W + 40.
Simplifying further:
4x + 4x = 40.
8x = 40.
Dividing both sides by 8, we find that:
x = 40/8.
x = 5 centimeters.
Therefore, the width of the border is 5 centimeters.