Determining Acceleration from Sinusoidal Velocity
Given a particle’s velocity function \(v(t) = \sin(t)\), where \(t\) is in seconds and \(v(t)\) is in meters per second, find the particle’s acceleration function \(a(t)\).
Comparing Local Minima in Polynomial Functions
Consider the function \( f(x) = x^4 – 2x^3 – 4x^2 \). This function has three critical points, with two of them corresponding to local minima and one to a local maximum. Identify which of the two local minima is lower by determining their \(x\)-values and comparing \(f(x)\) at these points.
Maximum Height of a Vertically Thrown Object
A ball is thrown vertically upward with an initial speed of \(v_0 = 20\,m/s\). Given that the acceleration due to gravity is \(g = 9.8\,m/s^2\) (acting downwards), determine the maximum height reached by the ball. (Assume upward is positive and air resistance is negligible.)
Optimisation in a Piecewise Cost Function Model
A manufacturing company uses a piecewise function to model its average production cost per unit. The average cost per unit function is defined as follows: For production quantities \(x < 500\): \(f(x) = ax^2 + bx + c\). For production quantities \(x \ge 500\): \(f(x) = dx + e\). Recent data from production runs gives […]
Optimizing Revenue with a Piecewise Pricing Strategy
A company adopts a pricing strategy that depends on the price charged per unit. The revenue function, \(R(p)\), is defined piecewise as follows: – For \(0 \leq p \leq 50\): \(R(p) = p(150 – 2p)\). – For \(50 < p \leq 120\): \(R(p) = p(100 – 0.5p)\). Here, \(p\) (in dollars) is the unit price […]
Constant Acceleration Vector Kinematics in Motion Analysis
A drone is launched from the origin with an initial velocity vector \(\vec{u} = \langle u_x, u_y, u_z \rangle\) and is subject to a constant acceleration vector \(\vec{a} = \langle a_x, a_y, a_z \rangle\). The drone’s vertical position \(z(t)\) is given by \(z(t) = u_z t + \frac{1}{2}a_z t^2\), and it must remain below a […]
Part c
For the horizontal motion given by \(d(t) = u_h t – \frac{1}{2}a_h t^2\), determine the time at which the horizontal displacement is maximized and calculate the maximum displacement.