1.A pie is circular in shape. One slice of the pie has an area of 5 in2 and a radius of 5 in. Find the central angle for this slice of pie in radians, rounded to the nearest tenth.
-0.4 radians
-0.1 radians
-2.5 radians
-2.0 radians
2.For a particular circle and a sector of the circle, Albert estimates that the area of the sector is around one-fourth or one-fifth of the area of the whole circle. Which of the following are possible measures for the central angle? Select the two correct answers.
-2π/3
-π/5
-π/2
-2π/5
-π/6
area of whole circle = pi r^2 = 25 pi
we have 1/5 of that circle which is 2 pi radians around
2 pi /5 = 0.4 pi
=======================================
2 pi / 4 = pi / 2
or
2 pi / 5
#1. A = 1/2 r^2 θ
#2. 1/4 or 1/5 of 2π
1. To find the central angle for the slice of pie, we can use the formula for the area of a sector of a circle:
Area of sector = (central angle / 2π) * πr^2
Given that the area of the slice of pie is 5 in^2 and the radius is 5 in, we can substitute these values into the formula:
5 = (central angle / 2π) * π * 5^2
Simplifying, we get:
5 = (central angle / 2) * 25
Dividing both sides by 25:
0.2 = central angle / 2
Multiply both sides by 2:
0.4 = central angle
Rounded to the nearest tenth, the central angle for this slice of pie is 0.4 radians.
Therefore, the correct answer is: -0.4 radians.
2. Albert estimates that the area of the sector is around one-fourth or one-fifth of the area of the whole circle. This means that the central angle of the sector is one-fourth or one-fifth of the central angle of the whole circle.
Since the area of a sector is proportional to the central angle, and the area of the sector is one-fourth or one-fifth of the area of the whole circle, the central angle of the sector is also one-fourth or one-fifth of the central angle of the whole circle.
The possible measures for the central angle are:
-2π/3 and -2π/5
Therefore, the correct answers are: -2π/3 and -2π/5.
1. To find the central angle of the pie slice, we need to use the formula for the area of a circle sector:
Area of sector = (angle in radians) * (radius^2) / 2
Given that the area of the slice is 5 in² and the radius is 5 in, we can rearrange the formula to solve for the angle:
(angle in radians) = (2 * Area of sector) / (radius^2)
Substituting the given values, we have:
(angle in radians) = (2 * 5 in²) / (5 in)² = (10 in²) / (25 in²) = 0.4 radians
Rounded to the nearest tenth, the central angle for this slice of pie is 0.4 radians.
Therefore, none of the given answer options (which are all negative angles) are correct for this question.
2. To determine the possible measures for the central angle of the sector, we need to consider the given statements regarding the area.
If the area of the sector is one-fourth of the area of the whole circle, then the ratio of the areas is:
Area of sector / Area of circle = 1/4
We know that the formula for the area of a circle sector is:
Area of sector = (central angle) * (radius^2) / 2
Similarly, if the area of the sector is one-fifth of the area of the whole circle, then the ratio of the areas is:
Area of sector / Area of circle = 1/5
By substituting the corresponding formulas, we get:
(central angle) * (radius^2) / 2 * (π * radius^2) = 1/4
and
(central angle) * (radius^2) / 2 * (π * radius^2) = 1/5
Simplifying these equations, we obtain:
(central angle) * (radius^2) / 2 * (π * radius^2) = 1/4
(central angle) * π * (radius^4) = 1/2
and
(central angle) * (radius^2) / 2 * (π * radius^2) = 1/5
(central angle) * π * (radius^4) = 2/5
Since (π * radius^4) is positive, dividing both sides of the equations by (π * radius^4) gives us:
central angle = (1/2) / (π * radius^4)
central angle = (2/5) / (π * radius^4)
Therefore, none of the given answer options (-2π/3, -π/5, -π/2, -2π/5, -π/6) satisfy the conditions given in the question.