AP Calc related rates ladder and balloon study guide

A 10-foot ladder leans against a wall. Height?

Height = 10 feet, Distance = x feet. Use Pythagorean theorem: $\displaystyle 10^2 = x^2 + h^2$.

When ladder base moves away, what happens?

Height decreases as the ladder base moves away from the wall.

True or False: Ladder length changes in related rates problems.

False. The ladder length remains constant in these problems.

If ladder base moves at 3 ft/sec, height change?

Height decreases at a rate calculated using related rates: $\displaystyle \frac{dh}{dt} = -\frac{3h}{10}$.

Fill in the blank: The distance from the wall is ____.

x, where $\displaystyle x^2 + h^2 = L^2$ (L = constant ladder length).

Compare height vs. distance from wall.

Height decreases as distance increases; inversely related due to Pythagorean theorem.

Height of ladder against wall formula?

Use: $\displaystyle h^2 + x^2 = L^2$, where L is ladder length.

If ladder is 12 ft and base moves 2 ft away...

Calculate rate of height decrease using $\displaystyle \frac{dh}{dt}$. Solve for $\displaystyle h$ first.

What do we differentiate in ladder problems?

Differentiate the Pythagorean theorem: $\displaystyle 2h \frac{dh}{dt} + 2x \frac{dx}{dt} = 0$.

How to find $\displaystyle \frac{dx}{dt}$?

Isolate $\displaystyle \frac{dx}{dt}$ in $\displaystyle 2h \frac{dh}{dt} + 2x \frac{dx}{dt} = 0$.

True or False: Ladder's position affects its height.

True. As the ladder moves, its height changes based on distance from the wall.

Initial height at 5 ft, base moves 1 ft/sec. Rate?

Calculate $\displaystyle \frac{dh}{dt}$ using derivative: $\displaystyle h$ decreases as distance $\displaystyle x$ increases.

What happens when ladder is vertical?

Height = ladder length, base distance = 0. Rate of height change is undefined.

Find height if base is 4 ft from wall.

Use $\displaystyle h^2 + 4^2 = 10^2$. Solve for $\displaystyle h$: $\displaystyle h = 8$ ft.

Factor affecting height decrease rate?

Distance from the wall (x) and current height (h) are key factors.

Determine effect on height if base moves faster.

Height decreases faster as base moves away at a greater rate.

A balloon rises at 3 ft/sec.

If the balloon is at a height of 10 ft, the rate of change of the height is 3 ft/sec.

True or False: The height of the balloon affects its distance from the observer.

True - As the balloon rises, its distance from the observer changes due to the height.

The observer is 100 ft away from the balloon's path.

If the balloon rises at 5 ft/sec, use the Pythagorean theorem to find the rate of distance change from the observer.

When is the distance from the balloon to the observer shortest?

When the balloon is directly above the observer, at the observer's horizontal distance.

If a balloon is at height h, what is the distance d?

Use the formula $\displaystyle d = \sqrt{(100)^2 + h^2}$. Find rates with respect to time.

Calculate the rate of distance change.

Distance $\displaystyle d = \sqrt{x^2 + h^2}$. Use implicit differentiation: $\displaystyle \\frac{dd}{dt} = \\frac{1}{2d}(2h\\frac{dh}{dt})$.

A balloon is 15 ft high, 75 ft away.

Differentiate to find the rate of distance from the observer as it rises 2 ft/sec.

If h = 20 ft, what is d?

Calculate: $\displaystyle d = \sqrt{(100)^2 + (20)^2} = \sqrt{10000 + 400} = \sqrt{10400} \approx 102 ft.$

As the balloon rises, what happens to d?

It increases; the height increase leads to a greater distance from the observer.

Rate of height increase is 4 ft/sec.

If h = 30 ft, find $\displaystyle \\frac{dd}{dt}$ using the related rates formula from previous cards.

Fill in the blank: The observer's distance is a _____ function of the balloon's height.

Increasing - as height increases, distance from observer increases.

The balloon rises at 1 ft/sec, observer 50 ft away.

At height 10 ft, find $\displaystyle \\frac{dd}{dt}$: Apply Pythagorean theorem and differentiate.

True or False: Height change does not affect horizontal distance.

False - Horizontal distance remains constant, but total distance changes with height.

A balloon is released from 5 ft.

If it rises at 3 ft/sec, what is the height after 10 seconds? Answer: 35 ft.

If h = 50 ft, what is the approximate rate of change of distance?

Use $\displaystyle d = \sqrt{(100)^2 + (50)^2}$, differentiate to find $\displaystyle \\frac{dd}{dt}$.

A balloon rises at 5 ft/sec and is 40 ft high.

Calculate the distance from the observer 30 ft away. Use Pythagorean theorem: $\displaystyle d = \\sqrt{(40^2 + 30^2)} = \\sqrt{1600 + 900} = \\sqrt{2500} = 50$ ft.