Math for - Functions, continuation

  1. Drone Battery Management

    An autonomous delivery drone's battery consumption (in Watt-hours, Wh) for a flight is modelled by the function $C(d) = 10d + 50$, where $d$ is the distance travelled in kilometers (km).

    1. Define a new function $R(B, d)$ that calculates the remaining battery capacity (in Wh) if the drone starts with an initial battery capacity $B$ and travels a distance $d$.

    2. If a drone starts with an initial battery capacity of 500 Wh and travels 15 km, what is its remaining battery capacity?

  2. Kinetic Energy Calculation

    The kinetic energy ($KE$) of a moving object in Joules (J) is given by the function $KE(m, v) = \frac{1}{2}mv^2$, where $m$ is the mass in kilograms (kg) and $v$ is the velocity in meters per second (m/s).

    Calculate the kinetic energy of a 0.5 kg projectile moving at a velocity of 20 m/s.

  3. Manufacturing Cost Analysis

    In a manufacturing process, the cost to produce $x$ units of a specific electronic component is given by the function $C_1(x) = 5x + 100$ (in dollars). Each final assembled product requires 3 of these components.

    1. Define a function $N(p)$ that determines the total number of components needed to produce $p$ final products.

    2. Define a new function $C_{total}(p)$ that calculates the total cost to produce all the components needed for $p$ final products. This function should use (or "call") both $N(p)$ and $C_1(x)$.

    3. Calculate the total cost if 50 final products are to be manufactured.

  4. Signal Processing Stages

    A signal processing system modifies an input signal $s$ in two sequential stages. The first stage, $A_1$, amplifies the signal: $A_1(s) = 3s$. The second stage, $A_2$, adds an offset to the amplified signal ($s_{amp}$): $A_2(s_{amp}) = s_{amp} + 5$.

    1. If the input signal is $s=4$ units, what is the signal value after the first stage, $A_1(s)$?

    2. Using the output from the first stage as the input to the second stage, what is the final signal value after the second stage, $A_2(s_{amp})$?

    3. Define a single function $T(s)$ that represents the entire two-stage transformation from the initial input $s$ to the final output.

  5. Solar Panel Efficiency Scenario

    The power output density $P_{out}$ (in W/m$^2$) of a specific type of solar panel is modeled by the function $P_{out}(T, I) = k \cdot I \cdot (1 - 0.005(T - 25))$, where:

    • $T$ is the ambient temperature in Celsius ($^\circ C$).
    • $I$ is the solar irradiance in Watts per square meter (W/m$^2$).
    • $k$ is a constant efficiency factor for the panel, $k=0.2$.

    1. Calculate the power output density if the temperature is $T = 30^\circ C$ and the solar irradiance is $I = 800$ W/m$^2$.

    2. What if the temperature increases to $T = 35^\circ C$, while the irradiance $I$ remains $800$ W/m$^2$? Calculate the new power output density and determine how it has changed compared to the result in part (a).

    3. What if (starting from the conditions in part (a)) the irradiance drops to $I = 600$ W/m$^2$, while the temperature $T$ remains $30^\circ C$? Calculate the new power output density and determine how it has changed compared to the result in part (a).