Related Rates · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

How linked quantities change together

Mathematics · Calculus & Analysis · ages 18-19
Name ______________________   Date ____________

  1. The radius of a circle grows at 3 cm per second. When the radius is 8 cm, how fast is the area growing?

    • 48 pi square cm per second
    • 24 pi square cm per second
    • 16 pi square cm per second
    • 6 pi square cm per second

  2. What is the first move when you meet a related rates problem?

    • Write the equation that links the changing quantities
    • Substitute the instant values into the formula
    • Divide one rate by the other

  3. You should substitute r equals 5 into the volume formula before differentiating.

    Circle one:   True   False

  4. A circle has radius 8 cm, growing at 3 cm per second. How fast is its area growing?

    • 16 pi square cm per second
    • 48 pi square cm per second
    • 6 pi square cm per second

  5. The edge of a growing cube is 4 cm and increasing at 0.5 cm per second. How fast is the volume growing, in cubic cm per second?

    Answer: ______________

  6. Air enters a balloon at 100 cubic cm per second and the radius is 5 cm. Which equation gives the radius rate?

    • 100 equals 8 pi times 5 times dr dt
    • 100 equals 4 pi times 25 times dr dt
    • 100 equals 4 thirds pi times 125 times dr dt

  7. A 10 m ladder slides down a wall. The foot is 6 m out and sliding at 2 m per second. How fast is the top moving, in m per second?

    Answer: ______________

  8. A 10 metre ladder slides down a wall. The foot is 6 m from the wall and sliding out at 2 m per second. How fast is the top moving?

    • 1.5 m per second downwards
    • 2 m per second downwards
    • 1.5 m per second upwards
    • 2.5 m per second downwards

  9. Two quantities satisfy x squared plus y squared equals 25. When x is 3 and y is 4, dx dt is 4 units per second. What is dy dt, in units per second?

    Answer: ______________

  10. Two quantities satisfy x squared plus y squared equals 25. When x is 3 and y is 4, dx dt is 4 units per second. What is dy dt in units per second?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

How linked quantities change together W1-mt_3XTIPt8Vmf-s1

  1. 48 pi square cm per second · Area is pi r squared, so dA dt is 2 pi r dr dt, which is 2 pi times 8 times 3, or 48 pi.
  2. Write the equation that links the changing quantities · The linking equation is what lets you connect the two rates.
  3. False · Substitute only after differentiating, or the radius freezes into a constant.
  4. 48 pi square cm per second · dA dt equals 2 pi times 8 times 3, which is 48 pi square cm per second.
  5. 24 · Volume is e cubed, so dV dt is 3 e squared de dt, which is 3 times 16 times 0.5, or 24.
  6. 100 equals 4 pi times 25 times dr dt · Differentiating first gives 100 equals 4 pi times 25 times dr dt.
  7. -1.5 · The top slides down at 1.5 m per second, so the signed rate is negative 1.5.
  8. 1.5 m per second downwards · From x squared plus y squared equals 100, y is 8, and dy dt equals minus x over y times dx dt, which is minus 1.5.
  9. -3 · Differentiating gives 2 x dx dt plus 2 y dy dt equals 0, so dy dt equals negative 3.
  10. -3 · Differentiating gives 2 x dx dt plus 2 y dy dt equals 0, so dy dt equals minus 3 over 4 times 4, or minus 3.
Worksheet · LightMySky