1) Find correct to six decimal places root of the equation cos(x)= x for xE[0, pi/2] using Newton's Method.
2) A triangle has two constant lengths of 10 cm and 15 cm. The angle between two constant sides increases at a rate of 9 deg/min. Find the rate of increase of the third side of the triangle when the angle between the two constant sides is 60 deg.
HELP WILL BE HUMBLY APPRECIATED!!!
Hukapan ponnaya
1) To find the root of the equation cos(x) = x using Newton's Method, we need to follow these steps:
Step 1: Choose an initial guess x₀, which lies between 0 and π/2 (since the given range is [0, π/2]). Let's say x₀ = 0.5.
Step 2: Calculate the derivative of the function f(x) = cos(x) - x, which is f'(x) = -sin(x) - 1.
Step 3: Use the Newton's Method iteration formula: xn+1 = xn - (f(xn) / f'(xn)).
Step 4: Repeat Step 3 until you reach a desired level of accuracy.
Let's start with the initial guess x₀ = 0.5:
x₁ = x₀ - (cos(0.5) - 0.5) / (-sin(0.5) - 1)
Plug in the values and calculate x₁. This will be your new approximation.
Repeat this process until you reach the desired level of accuracy. In this case, we want the answer correct to six decimal places, so keep iterating until the first six decimal places are the same for two consecutive approximations.
2) To find the rate of increase of the third side of a triangle when the angle between two constant sides is 60 degrees, we'll use the law of cosines and differentiate with respect to time.
The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, we have:
c² = a² + b² - 2ab * cos(C)
Let's denote the third side of the triangle as c and the angle between the two constant sides as C. Given that a = 10 cm, b = 15 cm, and dC/dt = 9 deg/min, we need to find dc/dt when C = 60 degrees.
Differentiating the law of cosines with respect to time, we get:
2c * dc/dt = -2ab * sin(C) * dC/dt
Plugging in the values, we have:
2c * dc/dt = -2 * 10 * 15 * sin(60) * 9
Simplifying further, we get:
2c * dc/dt = -300 * (√3/2) * 9
Now, solve for dc/dt:
dc/dt = -300 * (√3/2) * 9 / (2c)
Substitute c = 15 (since it is given as a constant length) and calculate dc/dt. The result will be the rate of increase of the third side when the angle between the two constant sides is 60 degrees.