A sector of a circle of radius is 7cm substending an angle of 27% at all he center of the circle is used to form a cone.
A. Find the base radius of the cone.
B. Calculate the area of base of the cone
If the sector subtends an angle θ, then the arc length subtended is 7θ
That will be the circumference of the base of the cone, so its radius will be r = 7θ/2π.
Then, as usual, the area is πr^2
If you make a sketch you will realize that the major arc length of the sector will be the circumference of the base of the the cone
circumference of original circle = 14π cm
So the length of the arc for your cone = (333/360)(14π) = 259/20 π cm
Then for your base of the cone:
2π r = 259/20 π
2r = 259/20
r = 259/40 cm or 6.475 cm
Now you have the radius of the base, just use your area formula.
Go with oobleck, I used the major sector to form the cone,
it should be the minor arc.
works for either arc. I also was ready to assume the major arc, since the angle subtended is so small.
To find the base radius of the cone, we can use the fact that the circumference of the base of the cone is equal to the length of the sector of the circle.
A. Finding the base radius of the cone:
1. The length of the sector of the circle can be found using the formula: L = 2πr(θ/360), where L is the length, r is the radius of the sector, and θ is the angle in degrees.
In this case, the length of the sector is given as 7 cm, and the angle is 27% of 360 degrees (since the angle is given as a percentage). So, θ = (27/100) * 360 = 97.2 degrees.
Substituting the values into the formula, we get: 7 = 2πr(97.2/360).
2. Simplifying the equation, we have: r = (7 * 360) / (2π * 97.2).
3. Evaluating the expression, we find: r ≈ 4.88 cm.
Therefore, the base radius of the cone is approximately 4.88 cm.
B. Calculating the area of the base of the cone:
1. The area of the base of the cone can be found using the formula: A = πr^2, where A is the area and r is the radius of the base of the cone.
From the previous step, we found that the base radius of the cone is approximately 4.88 cm.
2. Substituting the value into the formula, we get: A = π * (4.88)^2.
3. Evaluating the expression, we find: A ≈ 75.5 cm^2.
Therefore, the area of the base of the cone is approximately 75.5 cm^2.