Calculus I · 2B · lesson
Pythagorean Related Rates
Learn pythagorean related rates with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Related ratesWhat this section is building
Learn pythagorean related rates with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
The picture supplies a constraint; implicit differentiation transmits known rates through that constraint to the unknown rate.
Draw and label first, write one relationship, differentiate with time, then substitute the snapshot measurements and rates.
Substituting numerical dimensions before differentiating and thereby erasing the very change the problem asks about.
Learning objectives
Solve ladder, separation, and distance problems using the Pythagorean theorem.
Distances Connected by a Right Triangle
Before the formulas
In Pythagorean Related Rates, distinguish a quantity from its rate and a distance from a component of that distance. Many errors come from assigning one symbol to two different lengths or from confusing the speed of an object with the speed of a shadow tip or line of sight.
Use a variable table if the diagram is crowded: quantity, meaning, units, known value, known rate. The table makes substitutions deliberate and prevents a snapshot value from being mistaken for a constant throughout the motion.
Read this graph as text
Sliding ladder geometry. The wall, ground, and ladder form a right triangle. The fixed ladder length gives x 2+y 2=L 2 , while dx/dt and dy/dt describe the horizontal and vertical endpoint speeds. As the foot moves away from the wall, x increases. Because the ladder length stays fixed, the top must move downward, so y decreases. Differentiating x 2+y 2=L 2 gives 2x dx/dt+2y dy/dt=0 , which already predicts opposite signs.
Every relationship in sliding ladder geometry is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: The diagram should support variable selection and sign reasoning before any numbers are used. It must show that L is constant while x and y vary. Arrows make the direction conventions explicit.
The wall, ground, and ladder form a right triangle. The fixed ladder length gives x 2+y 2=L 2, while dx/dt and dy/dt describe the horizontal and vertical endpoint speeds.
Right triangles connect rates along different directions
Whenever two changing perpendicular distances and a connecting distance appear, the Pythagorean theorem is the natural relationship. Differentiating gives , which weights each rate by the current geometry.
The snapshot lengths matter because the same horizontal speed can produce different changes in the diagonal depending on the triangle's shape at that moment.
Right-triangle models appear whenever two perpendicular distances determine a third: ladders, aircraft separation, radar range, and objects moving on perpendicular roads. The Pythagorean theorem supplies the constraint, while signs on the rates record the directions of motion.
A length can be positive while its derivative is negative. For example, the height of a ladder remains positive even as it decreases. Confusing a quantity with its rate is one of the quickest ways to produce a physically impossible answer.
Sliding ladder
A -foot ladder leans against a wall. The bottom slides away from the wall at ft/s. How fast is the top sliding down when the bottom is feet from the wall?
Worked solution
Write a real attempt before opening the supplied answer.
Two vehicles moving on perpendicular roads
If one vehicle is miles east of an intersection and another is miles north, their separation satisfies
Differentiating gives
Signs of and must reflect whether each vehicle moves toward or away from the intersection.
The constant hypotenuse in a ladder problem has derivative zero. Writing is not wrong, but . More often, students mistakenly insert a nonzero rate for a length the problem says is fixed.
Two aircraft tracked from a control station
Aircraft A is km east of a station and moving east at km/h. Aircraft B is km north and moving south at km/h. Their separation satisfies
At the instant described, , , and . Hence
so
Despite both aircraft moving rapidly, their separation is decreasing only at km/h because the two directional effects nearly cancel.
app-aircraft-01Aircraft coordinates are and km with rates , . Find separation rate.
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