POD week 16Problem 1

By: Nora R.

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)

Strategy

o Using skills I know from this lesson

Area of Semi-Circleo 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*1/2~7.9o So the area of the semi circle is approximately 7.9 in.²

4 1/2

(to find area of full circle.)

Missing Side Lengtho 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)

Missing Side Length (Continued)o Then I drew out an equation- 4.5²+b²=7.5²o I did this equation because it’s the Pythagorean theorem

using the side lengths from the triangle in the heart.

4.5²+b²=7.5²20.25+b²=56.25-20.25 -20.25 b²=36

36= 6

b=6 in.²Missing side length is 6 inches.

Area of Triangleo 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= 13.5o So the area of the triangle is 13.5 in²

Area of Half Hearto 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 in²Area of triangle: +13.5 in²

21.4 in²(Total area of half heart)

Area of Whole Sheet of Papero 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

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

Answer

o My final answer is the measurement of one side length of the square piece of paper is:

approximately 12.2 in²

Any questions?