Miranda enlarged a picture twice as shown below, each time using a scale factor of 3.
A rectangle with length 6 inches and width 4 inches is enlarged twice.
Which statements apply to the enlargements? Select three options.
The area of the first enlargement is 72 square inches.
The area of the second enlargement is 1,944 square inches.
The area of the second enlargement is (3 squared) squared times the original area.
The area of the second enlargement is 3 times the area of the first enlargement.
The ratio of the area of the first enlargement to the area of the original equals the square of the scale factor.
But to get the correct area after one enlargement, try
(6*3) * (4*3) = (6*4)*3^2
The statements that apply to the enlargements are:
1. The area of the second enlargement is 1,944 square inches.
2. The area of the second enlargement is (3 squared) squared times the original area.
3. The ratio of the area of the first enlargement to the area of the original equals the square of the scale factor.
To solve this problem, we need to understand the concept of enlargements using scale factors.
The scale factor represents the ratio of the lengths of corresponding sides in the original figure and the enlarged figure.
In this case, the scale factor is 3, which means every length in the original figure is multiplied by 3 to get the corresponding length in the enlarged figure.
Let's find the dimensions of the first enlargement:
Length = 6 inches * 3 = 18 inches
Width = 4 inches * 3 = 12 inches
The area of the first enlargement is calculated by multiplying the length and width:
Area = 18 inches * 12 inches = 216 square inches
Now, let's find the dimensions of the second enlargement:
Length = 18 inches * 3 = 54 inches
Width = 12 inches * 3 = 36 inches
The area of the second enlargement is calculated by multiplying the length and width:
Area = 54 inches * 36 inches = 1,944 square inches
Now, let's check the statements:
1. The area of the first enlargement is 72 square inches.
This statement is false. The area of the first enlargement is 216 square inches, not 72.
2. The area of the second enlargement is 1,944 square inches.
This statement is true. The area of the second enlargement is indeed 1,944 square inches.
3. The area of the second enlargement is (3 squared) squared times the original area.
This statement is true. (3^2)^2 is equal to 9^2, which is 81. So, the statement can be written as "The area of the second enlargement is 81 times the original area," which is true.
4. The area of the second enlargement is 3 times the area of the first enlargement.
This statement is false. The area of the first enlargement is 216 square inches, while the area of the second enlargement is 1,944 square inches. Therefore, the area of the second enlargement is not 3 times the area of the first enlargement.
5. The ratio of the area of the first enlargement to the area of the original equals the square of the scale factor.
This statement is true. The scale factor is 3, and the square of 3 is 9. The ratio of the area of the first enlargement (216 square inches) to the area of the original (24 square inches) is indeed 9.
Therefore, the correct statements are:
- The area of the second enlargement is 1,944 square inches.
- The area of the second enlargement is 81 times the original area.
- The ratio of the area of the first enlargement to the area of the original equals the square of the scale factor.
To start you out:
(6 * 4) * 3 = ?