Download POD week 16 Problem 1

POD week 16
Problem 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-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
(to find area of
1/2∏r² (using pi as 3.14)
full circle.)
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
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)
7 1/2
(c)
(b)
?
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
-20.25
b²=36
36= 6
b=6 in.²
Missing side length is 6 inches.
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
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.
Area of semi-circle: 7.9 in²
Area of triangle: +13.5 in²
21.4 in²
(Total area of 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 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.
Area of half heart: 21.4 in²
*7
149.8 in²
o So the area of the whole sheet of paper is 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?