Final answer:
To find the length of the arc of the curve from point P to point Q, use the arc length formula s = rθ. Calculate the radius of the circle using the distance formula and find the angle using the inverse tangent. Finally, plug the values into the arc length formula to get the answer.
Step-by-step explanation:
To find the length of the arc of the curve from point P to point Q, we can use the arc length formula. The formula is given by s = rθ, where s is the arc length, r is the radius of the circle, and θ is the angle subtended by the arc in radians.
In this case, we are given the coordinates of points P(1, 5) and Q(8, 8). We can find the radius of the circle by using the distance formula between the two points. The distance is √((8-1)^2 + (8-5)^2) = √(49 + 9) = √58.
The angle can be found using the equation tan(θ) = (y - 4) / x^2. Plugging in the coordinates of point Q, we get tan(θ) = (8 - 4) / 8^2 = 4/64 = 1/16. Taking the inverse tangent of both sides, we get θ ≈ 3.57 radians.
Now we can use the arc length formula: s = rθ = √58 * 3.57 ≈ 12 units. Therefore, the length of the arc from point P to point Q is approximately 12 units.