You are making a card for a friend that looks like a heart shape. Solve for the total area of the shape. Use 3.14 as pi. Round the answer to the nearest hundredth. (1 point) Responses 16.56 in.2 16.56 in. squared 7.14 in.2 7.14 in. squared 5.57 in.2 5.57 in. squared 10.28 in.2

Question

inside card 2 inch

Answer 1

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.

1. 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).

2. 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: $$2\times 6.28=12.56{\text{ in}}^2$$

3. 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: $$A=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times 4\times 2=4{\text{ in}}^2$$

4. 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."