I am trying to help my child out with a problem but yeah I am finding it veyr hard.
Gabe is spending two weeks on an archaeological dig. He finds a fragment of a circular plate that his leader thinks may be valuable. The arc length of the fragment is about 1/6 the circumference of the original complete plate and measures 1.65 inches. A similar plate found earlier has a diameter of 3.14 inches.
Write an an expression for the radius, r, of the earlier plate.
r=2π
r=π/4
r=π/3
r=π/2
What is the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6 the length of the circumference of a circle?
θ=2π/6
θ=2π/5
θ=2π/4
θ=2π/3
Write an expression for the arc length, S, intercepted by this central angle
Could Gabe's plate be a matching plate? Explain.
Write an an expression for the radius, r, of the earlier plate.
The earlier plate has a diameter of 3.14 inches, approximately equal to π inches.
The radius equals half the diameter.
What is the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6 the length of the circumference of a circle?
The central angle of a complete circle is 360°, or 2π radians.
So the central angle of an arc equal to 1/6 of the circumference is therefore 1/6th of the complete circle. Therefore the central angle is (1/6)*2π.
Write an expression for the arc length, S, intercepted by this central angle
The arc length of radius r and central angle θ is rθ. Since r and θ are known, solve for arc length.
Could Gabe's plate be a matching plate? Explain.
It is most likely a matching plate, because the 1/6th of the arc of the previous plate has an arc length of 3.14*π/6=1.643, well within the accuracy expected of the later plate.
To help your child with this problem, let's break it down step by step.
1. Calculate the radius, r, of the earlier plate:
- The diameter of the earlier plate is given as 3.14 inches.
- Since the diameter is twice the radius, we can divide it by 2 to find the radius.
- So, the expression for the radius, r, would be r = 3.14/2 = 1.57 inches.
2. Determine the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6 the length of the circumference of a circle:
- The full circumference of a circle is given by 2πr, where r is the radius of the circle.
- Since the arc length of the fragment is about 1/6 of the circumference, we can calculate the measure of the central angle using the formula θ = (1/6) * 2π.
- Simplifying it, we get θ = 2π/6 = π/3 radians.
3. Write an expression for the arc length, S, intercepted by this central angle:
- The formula for arc length, S, is given by S = θr, where θ is the central angle in radians and r is the radius.
- Plugging in the values, S = (π/3) * 1.57 = 1.57π/3 inches.
4. Determine if Gabe's plate is a matching plate:
- Since the fragment's arc length is 1.65 inches, we can compare it to the arc length we just calculated, which is 1.57π/3 inches.
- If the two lengths match, then Gabe's plate could be a matching plate.
- Comparing them, we can say that Gabe's plate is not a matching plate because the calculated arc length (1.57π/3) is different from the given arc length (1.65).
By following these steps, you should be able to answer the questions and help your child understand the solution as well.