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In k(O,r), the length of minor arc AB is 8 in and the area of triangle AOB is 10in^2. Find m

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Explanation:

We can start by using the formula for the area of a sector of a circle, which is:

A = (θ/360)πr^2

where A is the area of the sector, θ is the central angle of the sector (in degrees), and r is the radius of the circle.

Since the length of minor arc AB is 8 inches, and the circumference of the circle is 2πr, we can find the central angle of the sector by using the formula:

θ = (8/2πr) × 360

Simplifying this expression, we get:

θ = 1440/r

Next, we can use the formula for the area of a triangle, which is:

A = (1/2)bh

where A is the area of the triangle, b is the base of the triangle, and h is the height of the triangle.

In triangle AOB, the base is the length of minor arc AB (which we know is 8 inches), and the height is the distance from O to AB (which we'll call h). We can use the sine function to find h, since we have the angle θ/2 and the length of the radius r:

sin(θ/2) = h/r

h = r sin(θ/2)

Substituting the expression we found for θ earlier, we get:

h = r sin(720/r)

Now we can substitute these expressions for θ and h into the formula for the area of the sector:

10 = (1440/360)πr^2

Simplifying this expression, we get:

r^2 = 10/(π/4) = 40/π

And substituting this expression for r into the expression we found for h:

h = r sin(720/r) = (2/π)√(40/π) ≈ 1.85

Finally, we can use the formula for the area of the triangle:

10 = (1/2)(8)(h)

Solving for h, we get:

h = 5/2

Therefore, the height of the triangle is 5/2 inches, and the length of the minor arc AB is 8 inches. To find the length of the major arc ACB (which we'll call L), we can use the formula:

L = 2πr(θ/360)

Substituting the expressions we found for r and θ, we get:

L = 16π/√(40π) ≈ 12.6

Therefore, the length of the major arc ACB is approximately 12.6 inches, and the value of m is:

m = L + AB = 12.6 + 8 = 20.6 inches.

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