the perimeter of the starfish is 30 centimeters. This original picture is to be enlarged in both dimensions to an image that is similar to the original picture. If the area of the enlarged image is 9 times the area of the original picture, what is the perimeter in centimeters of the starfish in the enlarged image?
area = k * length ratio^2
sqrt (9) = 3 times length ratio
3*30 = 90
in other words for similar figures, 3 times a length gives 9 times the area
Well, if the original picture is being enlarged in both dimensions to a similar image, that means all sides are being multiplied by the same factor. Let's call this factor "x". So, the area of the enlarged image will be x^2 times the area of the original picture.
We know that the area of the enlarged image is 9 times the area of the original picture. Therefore, we have the equation:
x^2 * (original area) = 9 * (original area)
Simplifying this equation, we find:
x^2 = 9
Taking the square root of both sides, we get:
x = 3
So, each side of the image in the enlarged picture is 3 times longer than in the original picture. Since the original perimeter is 30 centimeters, the perimeter of the enlarged image will be:
Perimeter of enlarged image = 3 * (original perimeter)
= 3 * 30
= 90 centimeters
Therefore, the perimeter of the starfish in the enlarged image would be 90 centimeters.
Let's assume the original perimeter of the starfish is "P" centimeters. You mentioned that the perimeter of the starfish is 30 centimeters, so P = 30 cm.
The area of the enlarged image is 9 times the area of the original picture. Therefore, the area of the enlarged image = 9 * (original area).
We're given that the dimensions are enlarged equally in both height and width. Let's assume the scale factor for enlargement is "k". So the dimensions of the enlarged image would be k times the dimensions of the original image.
Now, let's find the relationship between the areas of the original and enlarged images using the scale factor:
(original area) = (original length) * (original width)
(enlarged area) = (k * original length) * (k * original width) = k^2 * (original area)
Since the enlarged area is 9 times the original area, we can write:
9 * (original area) = k^2 * (original area)
(original area) cancels out, giving us:
9 = k^2
Taking the square root of both sides, we find:
k = 3
Now, we can find the dimensions of the enlarged image:
(enlarged length) = 3 * (original length)
(enlarged width) = 3 * (original width)
Since the dimensions are enlarged by a factor of 3, the perimeter of the enlarged image would also be enlarged by the same factor. Therefore, the perimeter of the starfish in the enlarged image would be:
(enlarged perimeter) = 3 * (original perimeter)
Substituting the value of the original perimeter:
(enlarged perimeter) = 3 * P
(enlarged perimeter) = 3 * 30 cm
(enlarged perimeter) = 90 cm
So, the perimeter of the starfish in the enlarged image would be 90 centimeters.
To find the perimeter of the starfish in the enlarged image, we first need to find the scale factor by which the image is being enlarged.
Let's assume that the original dimensions of the starfish are L (length) and W (width). The perimeter of the starfish is given as 30 centimeters.
Perimeter (P) = 2(L + W) = 30
Now, we need to find the scale factor by which the image is enlarged. Let's call it k.
The area of the enlarged image is given as 9 times the area of the original image. So we have:
Area of enlarged image = 9 * Area of original image
The area of a rectangle is given by multiplying the length and width. So we can write this as:
(kL) * (kW) = 9 * (L * W)
Simplifying, we get:
k^2 * L * W = 9 * L * W
Since the dimensions L and W are positive, we can divide both sides by L * W:
k^2 = 9
Taking the square root of both sides, we find:
k = 3
So, the scale factor of enlargement is 3. This means that the dimensions of the enlarged image are 3 times the dimensions of the original image.
Now, we can find the new perimeter of the enlarged image. Let's call it P'.
Since the dimensions of the enlarged image are 3 times the dimensions of the original image, we have:
Perimeter of enlarged image (P') = 2(3L + 3W) = 2 * 3(L + W) = 3 * (2(L + W)) = 3 * P
Therefore, the perimeter of the starfish in the enlarged image is 3 times the perimeter of the original starfish.
P' = 3 * P = 3 * 30 = 90 centimeters.
So, the perimeter of the starfish in the enlarged image is 90 centimeters.