To solve for the total area of a heart-shaped card, we can approximate it using the formula for the area of a circle and a triangle, since the heart shape can be thought of as two semicircles on top of a triangle.
-
Determine the dimensions:
- If the heart shape has an inner area of 2 inches, we can assume this refers to the radius of the semicircles. Hence, the radius \( r \) is 2 inches.
- The width of the heart shape would be \( 2r = 4 \) inches (the diameter of the semicircles).
-
Area of two semicircles:
- Area of a full circle: \( A = \pi r^2 \)
- So, the area of one semicircle: \[ \frac{1}{2} \times \pi r^2 = \frac{1}{2} \times 3.14 \times (2)^2 = \frac{1}{2} \times 3.14 \times 4 = 6.28 \text{ in}^2 \]
- Total area of two semicircles: \[ 2 \times 6.28 = 12.56 \text{ in}^2 \]
-
Area of the triangle:
- The height of the triangle can be estimated as equal to the radius of the semicircles (2 inches) since it sits directly below the two semicircles.
- The base of the triangle is the same as the width of the heart: 4 inches.
- Area of the triangle: \[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 2 = 4 \text{ in}^2 \]
-
Total area of the heart shape:
- Now we sum the areas of the two semicircles and the triangle: \[ \text{Total area} = \text{Area of semicircles} + \text{Area of triangle} = 12.56 + 4 = 16.56 \text{ in}^2 \]
Therefore, the total area of the heart-shaped card is 16.56 in², rounded to the nearest hundredth. The correct response would be "16.56 in. squared."