A circle with a radius of 8 centimeters. Two radii form a right angle. The area between them is shaded.
Question
What is the area of the shaded region, rounded to the nearest tenth?
Answer options with 5 options
A.
12.6 centimeters squared
B.
201 centimeters squared
C.
50.2 centimeters squared
D.
100.5 centimeters squared
E.
151 centimeters squared
To find the area of the shaded region, we need to find the area of the sector formed by the two radii and subtract the area of the triangle formed by the two radii and the hypotenuse.
The area of a sector can be found using the formula:
Area of sector = (angle/360) * π * r^2
Since the angle formed by the two radii is a right angle, the angle is 90 degrees. Plugging in the values into the formula:
Area of sector = (90/360) * π * 8^2 = (1/4) * π * 64 = 16π cm^2
The area of a triangle can be found using the formula:
Area of triangle = (1/2) * base * height
The base is the length of one of the radii, which is 8 cm. The height can be found using the Pythagorean theorem:
height^2 = hypotenuse^2 - base^2
height^2 = (8^2) - (8^2) = 64
height = √64 = 8 cm
Plugging in the values into the formula:
Area of triangle = (1/2) * 8 * 8 = 32 cm^2
Finally, we subtract the area of the triangle from the area of the sector to find the area of the shaded region:
Area of shaded region = Area of sector - Area of triangle = 16π cm^2 - 32 cm^2
Since we are asked to round to the nearest tenth, we can use an approximation for the value of π. Taking π ≈ 3.14, we can calculate:
Area of shaded region ≈ (16 * 3.14) cm^2 - 32 cm^2 ≈ 50.24 cm^2
Therefore, the area of the shaded region, rounded to the nearest tenth, is approximately 50.2 cm^2, which corresponds to option C.
What is the value of cos (4π/3)
To find the value of cos (4π/3), we can use the unit circle and the values of cosine for the special angles.
The angle 4π/3 is equivalent to 240 degrees.
In the unit circle, the coordinate of a point on the circle that corresponds to an angle θ has coordinates (cos θ, sin θ).
For an angle of 240 degrees or 4π/3, the cosine value is negative and given by the y-coordinate of the point on the unit circle.
In this case, the y-coordinate or sin θ is -1/2.
Therefore, cos (4π/3) = -1/2.
The graph shows triangle E F G on a coordinate plane. Point E is at (negative 4 comma 7). Point F is at (3 comma 7). Point G is at (3 comma negative 6).
Question
To the nearest tenth, what is the length of segment E G? Enter the answer in the box.
To find the length of segment EG, we can use the distance formula.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we have the coordinates of point E as (-4, 7) and the coordinates of point G as (3, -6).
Plugging the values into the distance formula:
d = √((3 - (-4))^2 + (-6 - 7)^2)
Simplifying:
d = √(7^2 + (-13)^2)
d = √(49 + 169)
d = √(218)
Rounding to the nearest tenth:
d ≈ 14.8
Therefore, the length of segment EG, rounded to the nearest tenth, is approximately 14.8 units.
on to answer the question.
Information
Sara's hair was 25 inches long. Three years later it was 43 inches long.
Question
What was the percent increase, to the nearest hundredth, of Sara's hair length? Enter the answer in the box.
To calculate the percent increase of Sara's hair length, we can use the formula:
Percent Increase = (Change in value / Original value) * 100
In this case, the original value is 25 inches and the new value is 43 inches. The change in value is the difference between the new value and the original value, which is 43 - 25 = 18 inches.
Plugging the values into the formula:
Percent Increase = (18 / 25) * 100
Calculating:
Percent Increase ≈ 72
Therefore, the percent increase of Sara's hair length, rounded to the nearest hundredth, is approximately 72%.