Use the logarithm tables to evaluate If T = 2 √G π , make G the subject of the formula. Hence, find the value of G to the nearest whole number when T=45, L=8 and π = 3.142.

a) To find the value of P, we first find the derivative of the function y with respect to x. A stationary value occurs when the first derivative is equal to zero. Step 1: Differentiate y = 2x^4 - Px^2 + 2x with respect to x. (dy)/(dx) = (d)/(dx)(2x^4 - Px^2 + 2x) = 8x^3 - 2Px + 2 Step 2: Set (dy)/(dx) = 0 and substitute x = -1 to solve for P. 8(-1)^3 - 2P(-1) + 2 = 0 -8 + 2P + 2 = 0 2P - 6 = 0 2P = 6 P = 3 The value of P is 3. b) The position of the particle is given by x = 5t^3 - 4t^2. (i) Velocity Step 1: Find the velocity function by differentiating the position function with respect to time. v(t) = (dx)/(dt) = (d)/(dt)(5t^3 - 4t^2) = 15t^2 - 8t Step 2: Calculate the velocity at t = 4 seconds. v(4) = 15(4)^2 - 8(4) v(4) = 15(16) - 32 v(4) = 240 - 32 v(4) = 208 The velocity after 4 seconds is 208 units/s. (ii) Acceleration Step 3: Find the acceleration function by differentiating the velocity function with respect to time. a(t) = (dv)/(dt) = (d)/(dt)(15t^2 - 8t) = 30t - 8 Step 4: Calculate the acceleration at t = 4 seconds. a(4) = 30(4) - 8 a(4) = 120 - 8 a(4) = 112 The acceleration after 4 seconds is 112 units/s^2. c) To evaluate the definite integral, we first find the antiderivative of the function and then apply the limits of integration. Step 1: Find the indefinite integral of 6x^2 - 4x + 2. (6x^2 - 4x + 2) \,dx = 6x^2+12+1 - 4x^1+11+1 + 2x^0+10+1 + C = (6x^3)/(3) - (4x^2)/(2) + 2x + C = 2x^3 - 2x^2 + 2x + C Step 2: Evaluate the definite integral from 0 to 2. _0^2 (6x^2 - 4x + 2) \,dx = [ 2x^3 - 2x^2 + 2x ]_0^2 = (2(2)^3 - 2(2)^2 + 2(2)) - (2(0)^3 - 2(0)^2 + 2(0)) = (2(8) - 2(4) + 4) - (0 - 0 + 0) = (16 - 8 + 4) - 0 = 8 + 4 = 12 The value of the integral is 12. Send me the next one 📸