Suppose a triangular silk scarf has both of its shortest sides 14 inches long. If the angle between these sides is 110º, what is the area of the scarf? Round your answer to the nearest tenth of a square inch. (Enter only the number.)
if the third side is 2x, then
x = 14 sin 55º, and the height is
h = 14 cos 55º,
so the area is hx = (14 cos55º)(14 sin55º) = 7 sin 110º
To find the area of the triangular silk scarf, we can use the formula for the area of a triangle, which is given by:
Area = (1/2) * base * height
In this case, the base of the triangle is one of the shortest sides of the scarf, which is 14 inches. To find the height, we need to use trigonometry.
Since we know the angle between the two shortest sides of the triangle is 110º, we can use the sine function to find the height. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the angle is the height of the triangle, and the hypotenuse is one of the shortest sides of the scarf.
Using the formula for the sine function:
sin(angle) = opposite / hypotenuse
We can rearrange the formula to solve for the height:
height = opposite = sin(angle) * hypotenuse
height = sin(110º) * 14 inches
Now, we can calculate the height of the triangle using the sine function:
height = sin(110º) * 14 inches
≈ 13.99 inches
Now, we can substitute the values of the base and height into the formula for the area of a triangle:
Area = (1/2) * base * height
= (1/2) * 14 inches * 13.99 inches
≈ 97.9 square inches
Therefore, the approximate area of the scarf is 97.9 square inches.