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.