Date post: | 19-Dec-2015 |
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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)
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