A cone is created from a paper circle with a 90° sector cut from it. The paper along the remaining circumference of the circle is the base of the cone. Find the radius of the base of the cone. Round to the nearest hundredth.
In Foundations of Art, students are cutting circular paper plates into sectors. If the diameter of the plate is 22 cm, determine the central angle necessary to create a sector with an area of 220 cm2. Round to nearest thousandth if necessary.
since you have neglected to mention the radius of the paper circle, it's hard to give a numeric answer.
Suffice it to say that since 1/4 of the paper has been removed, the radius of the cone's base is 3/4 the radius of the original paper circle.
To find the radius of the base of the cone, we first need to determine the radius of the original paper circle.
Since a 90° sector is cut from the circle, the remaining sector forms a right-angled triangle inside the circle. The hypotenuse of this triangle is equal to the radius of the circle.
Now, let's assume that the length of the hypotenuse (radius of the circle) is 'r'. The two sides of the right-angled triangle are both equal to the radius of the sector that was cut (since the triangle is an isosceles right triangle).
By the Pythagorean Theorem, we can find the value of 'r'.
The formula of the Pythagorean Theorem is:
c² = a² + b²,
where 'c' is the hypotenuse and 'a' and 'b' are the other two sides of the right-angled triangle.
In this case, 'c' is the radius of the circle (which we need to find), and 'a' and 'b' are the two equal sides of the right-angled triangle (the radius of the sector that was cut).
So, we have:
r² = a² + b²,
where 'r' is the radius of the circle and 'a' is the radius of the sector that was cut.
Since the radius of the sector is equal to the length of its arc (circumference of the sector) divided by 2π, we have:
a = length of the arc / 2π.
Since the length of the remaining circumference is equal to the original circumference minus the length of the arc, we have:
remaining circumference = original circumference - length of the arc.
The formula for the circumference of a circle is:
circumference = 2πr,
where 'r' is the radius of the circle.
In this case, the remaining circumference is equal to the circumference of the circle minus the length of the arc, so we have:
remaining circumference = 2πr - length of the arc.
Since the sector angle is 90° (π/2 radians), the length of the arc can be found using the formula:
length of the arc = (sector angle / 2π) * circumference.
Substituting the formula for the length of the arc into the formula for the remaining circumference, we have:
remaining circumference = 2πr - [(sector angle / 2π) * circumference].
Now, let's substitute the values into the formula and solve for 'r'.
sector angle = 90° = π/2 radians,
circumference = remaining circumference,
a = length of the arc / 2π = remaining circumference / (2π * 2π).
r² = a² + b²,
r² = [(remaining circumference / (2π * 2π))]² + (remaining circumference)²,
r² = [remaining circumference² / (4π²)] + remaining circumference²,
r² = (5/4π²) * remaining circumference²,
r = sqrt[(5/4π²) * remaining circumference²].
The radius of the base of the cone is approximately sqrt[(5/4π²) * remaining circumference²].
To find the radius of the base of the cone, we need to use the given information of a 90° sector cut from the paper circle.
First, let's understand the properties of a cone. A cone has a circular base and a pointed top called the apex. The distance from the apex to the base, passing through the center of the base, is called the height of the cone. The radius of the base is the distance from the center of the base to any point on its circumference.
Now, since we have a 90° sector cut from the circle, it means that the remaining circumference of the circle forms the base of the cone. Let's call this remaining circumference C. We can find C using the formula for the circumference of a circle.
The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.
In our case, since we want to find the radius of the cone's base, let's call it r'. The formula for the remaining circumference of the circle, which is the base of the cone, becomes C = 2πr'.
Now, we know that the 90° sector cut from the circle forms the curved surface of the cone. The length of this curved surface is equal to the circumference of the base of the cone (which is C).
The length of the curved surface is given by the formula S = πr, where r is the radius of the cone (which is r'). In our case, we're given that the length of the curved surface is equal to half the circumference of the base of the cone.
So, S = 0.5C.
Substituting the value of C from the earlier equation (C = 2πr'), we get:
0.5C = 0.5 * 2πr' = πr'.
Now, we know that the length of the curved surface (S) is also given by the formula S = πr.
Equating the two formulas for S, we have:
πr = πr'.
Canceling out π from both sides of the equation, we get:
r = r'.
Therefore, the radius of the base of the cone is equal to the radius of the paper circle that was used to create it.
Hence, the radius of the base of the cone (r') is the same as the radius of the original paper circle.
To find the radius of the base of the cone, we would need to know the radius of the original paper circle. Since it is not mentioned in the question, we cannot determine the exact value.