A fashion designer created a sketch of a square scarf. The design has one large triangle and two congruent smaller triangles. The shaded portion shows the part made from red silk. The sketch of the scarf has a scale of 5 inches = 3 feet. How much red silk does the fashion designer need to make the scarf?
An equilateral triangle within a square that is 5 inches wide. The base of the triangle is the same width as the square and the top of the triangle touches the top of the square.
CLEAR SUBMIT
2.25 ft2
2
.
25
ft
2
4.5 ft2
4
.
5
ft
2
6.25 ft2
6
.
25
ft
2
12.5 ft2
To find out how much red silk the fashion designer needs to make the scarf, we first need to calculate the area of the shaded portion.
The sketch of the scarf has a scale of 5 inches = 3 feet. Since the width of the square is 5 inches, the actual width of the square is (5 inches * 3 feet / 5 inches) = 3 feet.
The large triangle inside the square is an equilateral triangle. Since the base of the triangle is the same width as the square, the base of the triangle is also 3 feet.
To find the height of the triangle, we can use the formula h = (sqrt(3) * a) / 2, where a is the length of one side of the triangle (which is also 3 feet in this case).
Plugging in the values, h = (sqrt(3) * 3 feet) / 2 = (1.732 * 3 feet) / 2 = 2.598 feet.
Now, we can calculate the area of the large triangle using the formula area = (base * height) / 2 = (3 feet * 2.598 feet) / 2 = 7.794 square feet.
Since there are two congruent smaller triangles, the total area of the smaller triangles is 2 * 7.794 square feet = 15.588 square feet.
Therefore, the fashion designer needs 15.588 square feet of red silk to make the scarf.
The area of the shaded portion is equal to the area of the large triangle minus the combined area of the two smaller triangles.
To find the area of the large triangle, we need to find the length of its base and its height. Since the triangle is equilateral, the length of its base is equal to the width of the square, which is 5 inches.
To find the height of the triangle, we can use the Pythagorean theorem. The height is the hypotenuse of a right triangle with one side equal to half the length of the base (5/2 inches) and another side equal to the height of the square (also 5 inches).
Using the Pythagorean theorem:
height^2 = (5/2)^2 + 5^2
height^2 = 25/4 + 25
height^2 = 100/4 + 100/4
height^2 = 200/4
height^2 = 50
height = √50
The area of the large triangle is (1/2) * base * height = (1/2) * 5 * √50 = (5/2) * √50
To find the area of the two smaller triangles, we can divide the area of the large triangle by 2. Therefore, the area of each smaller triangle is (5/2) * √50 / 2 = (5/4) * √50
The total area of the shaded portion is then the area of the large triangle minus the combined area of the two smaller triangles:
(5/2) * √50 - 2 * (5/4) * √50 = (5/2) * √50 - (10/4) * √50 = (5/2 - 5/2) * √50 = 0 * √50 = 0
Therefore, the fashion designer does not need any red silk to make the scarf.
To determine the amount of red silk needed to make the scarf, we first need to find the area of the shaded portion.
The sketch of the scarf has a scale of 5 inches = 3 feet. This means that each inch in the sketch represents 3/5 feet in real life.
The square in the sketch is 5 inches wide, which corresponds to (5 * 3/5) = 3 feet in real life.
The triangle in the sketch is an equilateral triangle that is positioned within the square. Since the triangle is equilateral, all sides are equal. The base of the triangle is the same width as the square (3 feet), and the top of the triangle touches the top of the square.
To find the height of the triangle, we can use the formula for an equilateral triangle: height = (side length * sqrt(3)) / 2.
In this case, the side length of the triangle is 3 feet, so the height is (3 * sqrt(3)) / 2 feet.
The area of an equilateral triangle can be found using the formula: area = (side length^2 * sqrt(3)) / 4.
In this case, the area of the large triangle is (3 * (3 * sqrt(3)) / 4) square feet.
Since there are two congruent smaller triangles, the total area of the shaded portion made from red silk is 2 * (3 * (3 * sqrt(3)) / 4) square feet.
Simplifying the expression, we get (9 * sqrt(3)) / 2 square feet.
Approximating the value of sqrt(3) to 1.732, the area of the shaded portion is (9 * 1.732) / 2 = 15.588 / 2 = 7.794 square feet.
Therefore, the fashion designer needs approximately 7.794 square feet of red silk to make the scarf.