Wemazi

Topic 4.4

Identify the changing quantities

Builds on separating amount, average rate and instantaneous rate. 6 minutes.

Water in a can

Water pours into a can with a radius of 3 centimeters. The teal region shows the water, and the orange arrow shows its depth.

Move the time slider. The depth \(\cy{h}\) and the volume \(\cb{V}\) of the water grow, but the radius of the can stays 3 centimeters.

The depth and the volume change with time, so you write them as functions of time, \(\cy{h(t)}\) and \(\cb{V(t)}\). The radius is a fixed length, so you write it as the number 3.

Changing or fixed

Move the slider to compare the water with the faint dashed line, which marks the surface at an earlier time.

Between the two times, the depth \(\cy{h}\) and the volume \(\cb{V}\) change, but the radius \(r\) stays the same.

A changing quantity has a rate, such as \(\cy{\frac{dh}{dt}}\) or \(\cb{\frac{dV}{dt}}\). A fixed quantity such as \(r\) is a constant, so its rate is 0.

Notation

  1. Draw the scene and label every length, area, volume and angle in the problem.

  2. Write each changing quantity as a function of time \(t\).

    \[\cy{h(t)}, \quad \cb{V(t)}\]

  3. Write each fixed length as a number or a constant, because it has the same value at every time.

    \[\cq{r = 3}\]

  4. Write each given rate and each wanted rate as a derivative with respect to \(t\).

    \[\cb{\frac{dV}{dt}}, \quad \cy{\frac{dh}{dt}}\]

  5. Assume that every changing quantity is a differentiable function of \(t\), so that its rate exists.

When it fails

A problem can give the value of a changing quantity at one instant, such as a depth of 2 centimeters at this moment.

This value does not make the depth fixed, because three seconds later the depth is 3.5 centimeters. Write \(h(t)\) for the depth, and use the value 2 only at the moment that the question names.

Example 1: A kite

A kite flies 40 meters above a person’s hand. The wind carries it away horizontally at 2 meters per second. List the quantities that change and the quantities that are fixed.

  1. The kite stays 40 meters above the hand at every time, so the height is fixed.

    \[\cq{40}\]

  2. The wind carries the kite away, so the horizontal distance \(x\) changes with time.

    \[\cy{x(t)}\]

  3. The string becomes longer as \(x\) grows, so its length \(s\) changes too.

    \[\cy{s(t)}\]

  4. The given speed of 2 meters per second is the rate at which \(x\) changes.

    \[\cy{\frac{dx}{dt}} = 2\]

Example 2: A conical tank

Water pours into a tank shaped like an upside-down cone. The top of the tank has a radius of 3 meters, and the tank is 6 meters tall. List the quantities that change and the quantities that are fixed.

  1. The tank does not change shape, so its height and its top radius are fixed.

    \[\cq{6}, \quad \cq{3}\]

  2. A tempting wrong move: The tank has a radius of 3, so the radius \(r\) of the water surface is fixed and has no rate.

    \[\cr{r = 3}\]

    The red line of radius 3 at the water surface reaches past the tank wall, because 3 is the radius at the top. The tank narrows downward, so at a depth of 2 meters the water surface has a radius of only 1 meter.

  3. As water pours in, the depth, the radius of the water surface and the volume all grow, so each is a function of time.

    \[\cy{h(t)}, \quad \cy{r(t)}, \quad \cb{V(t)}\]

Practice

A Ferris wheel has a radius of 30 meters, and its center is 35 meters above the ground. A rider goes around the wheel. The rider’s height is \(y\), and the angle between the horizontal through the center and the line to the rider is \(\theta\). List the quantities that change and the quantities that are fixed.

For each quantity, ask whether it has the same value at every time. Drag the rider and see which marks move.

Solution

  1. The wheel does not change size and its center does not move, so these two lengths are fixed.

    \[\cq{30}, \quad \cq{35}\]

  2. The rider moves around the wheel, so the angle and the height change.

    \[\cy{\theta(t)}, \quad \cy{y(t)}\]

  3. Only \(\theta\) and \(y\) have rates.

Questions

A swimming pool is 20 meters long and 8 meters wide. Water flows in, so the depth \(h\) and the volume \(V\) of the water grow. A dashed outline marks the surface of the water at an earlier time.

  1. Which quantity in the picture is fixed?

    • The width, 8 meters
    • The depth h
    • The volume V
    • The time t
    Answer

    The width, 8 meters

  2. Between the earlier time and the later time, which quantity has the same value?

    • The area of the water’s surface
    • The mass of the water
    • The depth of the water
    • The volume of the water
    Answer

    The area of the water’s surface

  3. A triangle has a fixed base of 6 centimeters. Its height \(h\) grows with time, and its area is \(A = 3h\). Which quantities need a rate?

    • \(h\) and \(A\)
    • \(h\) only
    • The base, \(h\) and \(A\)
    • \(A\) only
    Answer

    \(h\) and \(A\)

Summary

Before you write any equation, you list every quantity in the problem and decide whether it changes with time or stays fixed.

A changing quantity has a rate, but a fixed length has the same value at every time, so its rate is 0.

A value given at one instant does not make a quantity fixed, so you sort the quantities before you substitute any numbers.

This is one of four skills in topic 4.4 of Which & Why: AP Calculus.