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 folded paper. The pattern consists of a right triangle with hypotenuse of length 7 1/2 inches and a semi-circle with a diameter of 4 1/2 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. (Use ð = 22/7)
To solve this problem, we need to find the side length of the square piece of paper that Jessica is using to create the card.
First, let's calculate the total area of the piece of paper that Jessica is using. We know that the entire heart will cover 2/7 of the square paper.
Let's assume the side length of the square paper is "x" inches. The area of the square paper is x^2 square inches.
Since the entire heart will cover 2/7 of the square paper, the area covered by the heart is (2/7) * x^2 square inches.
Next, let's find the area of the heart. The heart consists of a right triangle and a semi-circle.
The area of the right triangle is given by the formula (1/2) * base * height. In this case, the base is the hypotenuse of the right triangle, which is 7 1/2 inches, and the height is the other side of the right triangle. We can use the Pythagorean theorem to find the height.
According to the Pythagorean theorem, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. So, we have (height^2) + (base^2) = (hypotenuse^2). Substituting the values, we get (height^2) + (7.5^2) = (7.5^2). Solving this equation, we find the height of the triangle is 5 inches.
Thus, the area of the right triangle is (1/2) * 7.5 * 5 = 18.75 square inches.
Now, let's find the area of the semi-circle. The formula for the area of a semi-circle is (1/2) * pi * (radius^2), where pi is approximately 22/7 and the diameter of the semi-circle is given as 4 1/2 inches. Therefore, the radius of the semi-circle is (4.5/2) = 2.25 inches. Substituting these values, we get the area of the semi-circle as (1/2) * (22/7) * (2.25^2) = 7.96 square inches (rounded to two decimal places).
Now, let's add the areas of the right triangle and the semi-circle to find the total area of the heart: 18.75 + 7.96 = 26.71 square inches (rounded to two decimal places).
We know that the area covered by the heart is (2/7) * x^2 square inches. So, we have (2/7) * x^2 = 26.71.
To find the side length of the square paper, we need to solve for x. Multiplying both sides of the equation by 7/2, we have x^2 = (26.71 * 7/2).
Calculating the right-hand side of the equation, we get x^2 = 93.485.
Taking the square root of both sides, we find x = √(93.485) ≈ 9.67 inches.
Therefore, the side length of the piece of paper Jessica is using to create the card is approximately 9.67 inches to the nearest tenth.