Linear functions：練習問題

The linear function \( g \) is defined by \( g(x) = \frac{1}{2}x + 10 \). What is the value of \( g(20) \)?

A candle is 20 centimeters tall and burns at a constant rate of 0.5 centimeters per hour. The height \(h\) of the candle, in centimeters, after burning for \(t\) hours is given by the function \(h(t) = 20 - 0.5t\). What does the value 20 represent in this context?

The table below shows several values for a linear function \(g\). What is the value of \(g(10)\)?

\(x\): 2, 4, 6
\(g(x)\): 5, 11, 17

A linear function is defined by \(f(x) = -2x + 5\). What is the value of \(f(-3)\)?

In the \(xy\)-plane, the graph of a linear function \(f\) passes through the points \((0, 4)\) and \((2, 10)\). What is the value of \(f(5)\)?

The total cost \( C \), in dollars, to produce \( n \) items is given by the linear function \( C(n) = 15n + 2500 \). If the total cost was \( \$10,000 \), how many items were produced?

A water tank is being filled at a constant rate. After \( 3 \) hours, the tank contains \( 450 \) gallons of water. After \( 7 \) hours, the tank contains \( 850 \) gallons of water. Let \( W \) represent the amount of water in the tank in gallons and \( t \) represent the time in hours since the filling began.

Part a: Determine the linear function \( W(t) \) that models the amount of water in the tank.
Part b: What is the rate of change of the water volume in gallons per hour?
Part c: How much water was in the tank when the filling process started?

A delivery drone is programmed to monitor its battery percentage during a two-leg mission. The drone starts with a \(95\%\) battery charge. During the first leg, the drone travels \(10\) kilometers (km) without a load, losing battery at a constant rate of \(2\%\) per km. For the second leg, the drone carries a package and loses battery at a constant rate of \(3.5\%\) per km.

Part a: Calculate the remaining battery percentage after the first leg, and write a linear function \(B(x)\) that represents the battery percentage remaining after the drone travels an additional \(x\) km during the second leg.
Part b: Safety protocols require the drone to land with at least \(15\%\) battery remaining. What is the maximum distance, in km, the drone can travel during the second leg?
Part c: If the drone travels at a constant speed of \(30\) km/h during the second leg, express the battery percentage \(B\) as a linear function of time \(t\), where \(t\) is the time in minutes since the start of the second leg.