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. What is the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6

Question

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. 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?

Answer 1

I will be happy to critique your thinking.

Answer 2

To start, let's write an expression for the radius, r, of the earlier plate.

The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius.

Since the arc length of the fragment is 1/6 the circumference of the complete plate, we can say that the arc length of the fragment is (1/6) * C.

We are given that the arc length of the fragment is 1.65 inches, so we have the equation:

(1/6) * C = 1.65

To find C, we need the diameter of the complete plate. The diameter is twice the radius, so we have:

D = 2r

We are given the diameter of the complete plate as 3.14 inches, so we have the equation:

2r = 3.14

Now, we can substitute C and r into the first equation:

(1/6) * (2πr) = 1.65

Simplifying, we have:

(πr) / 3 = 1.65

Now, let's solve for r:

πr = 1.65 * 3

πr = 4.95

r = 4.95 / π

Therefore, the expression for the radius, r, of the earlier plate is r = (4.95 / π).

Now, let's move on to the second part of the question.

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 can be found using the formula:

θ = (arc length) / (radius)

In this case, the arc length is 1/6 the circumference of the complete plate, which is (1/6) * C.

We have already found the expression for the radius, r, as r = (4.95 / π).

Substituting these values into the formula, we have:

θ = ((1/6) * C) / r

Simplifying, we have:

θ = ((1/6) * (2πr)) / r

θ = π / 3 radians

Therefore, 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 is π / 3 radians.