1. POD week 16 Problem 1 By: Nora R.

2. Problem Jessica is creating a heart-shaped card to give to her valentine. She has folded a square piece of paper in half and traced the pattern onto one half of the paper. The pattern consists of a right triangle with hypotenuse of length 7 ½ inches and a semi-circle with a diameter of 4 ½ inches. The entire heart will cover 2/7 of the square paper. What is the side length, in inches, of the piece of paper Jessica is using to create the card? Express your answer as a decimal to the nearest tenth. (∏=3.14)

3. Strategy o Using skills I know from this lesson

4. Area of Semi-Circle o First I decided to find the area of the semi circle in the heart. (∏r²) but since this is only half of a circle my formula was 1/2∏r² (using pi as 3.14) o The diameter of the semi-circle is 4 ½ so I divided that by 2 to get the radius (2.25) o 2.25²*3.14*1/2~7.9 o So the area of the semi circle is approximately 7.9 in.² 4 1/2 (to find area of full circle.)

5. Missing Side Length o Next, I decided to find the missing side length in the triangle by using the Pythagorean Theorem (a²+b²=c²) o The side length of 4 ½ represents a, the missing side length is b, and the hypotenuse of 7 ½ represents c. ? 4 1/2 (a) (b) 7 1/2 (c)

6. Missing Side Length (Continued) o Then I drew out an equation- 4.5²+b²=7.5² o I did this equation because it’s the Pythagorean theorem using the side lengths from the triangle in the heart. 4.5²+b²=7.5² 20.25+b²=56.25 -20.25 b²=36 36= 6 b=6 in.² Missing side length is 6 inches.

7. Area of Triangle o Now I can find the area of the triangle by doing half of base times height. (1/2*b*h) o.5*6*4.5= 13.5 o So the area of the triangle is 13.5 in²

8. Area of Half Heart o After I found both areas from the semi-circle and the triangle, I added both areas together to get the total area of the half heart. o And since I know that the whole heart is 2/7 of the paper I know that half the heart equals 1/7. (since 1/7 is half of 2/7) Area of semi-circle: 7.9 in² Area of triangle: +13.5 in² 21.4 in² (Total area of half heart)

9. Area of Whole Sheet of Paper o Since I know that half the heart equals 1/7 I multiplied the area of the half heart by seven to get the total area of the whole square paper. o So the area of the whole sheet of paper is 149.8 in² Area of half heart: 21.4 in² *7 149.8 in²

10. Finding the Square Root o Finally I found the square root of the whole sheet of paper’s area to get the measure of one side length. 149.8= approximately 12.2 in²

11. Answer o My final answer is the measurement of one side length of the square piece of paper is: approximately 12.2 in²

12. Any questions?