A 13-foot ladder leans against a wall. The bottom slides away from… Step by Step

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The question

Question: A 13 ft ladder leans against a wall.

The bottom slides away at 2 ft/s.

How fast is the top sliding down when the bottom is 5 ft from the wall?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Draw and name

Draw it.

The ladder stays 13 ft long.

The bottom's distance from the wall, x, and the top's height, y, change.

The bottom slides away: x grows by 2 ft per second.

So dx/dt = 2.

dxdt = 2 ft/s

dx/dt means how fast x changes: the change each second.

dxdt: how fast x changes, per second

We want dy/dt at the moment when x = 5.

dydt = ?

2 Write the equation

The wall, the ground and the ladder make a right triangle.

Pythagoras: x squared plus y squared is 13 squared.

x² + y² = 13²

Careful!

Do not put x = 5 in yet!

x changes with time.

Numbers go in only after the derivative.

Careful: not 5² + y² = 13² first ✗

3 Take the derivative

Both x and y change with time.

The chain rule: x squared gives 2x times dx/dt.

y squared gives 2y times dy/dt.

13 squared is a number that never changes, so its derivative is 0.

2x dxdt + 2y dydt = 0

4 Put the numbers in

First the missing side at that moment: y.

Pythagoras with x = 5.

5² + y² = 13²

13 squared is 169, 5 squared is 25.

Take away: 144.

y² = 169 − 25 = 144

So y = 12.

A length is positive.

y = √144 = 12

Now put in x = 5, y = 12, and dx/dt = 2.

2(5)(2) + 2(12)dydt = 0

Multiply out: 2 times 5 times 2 is 20, and 2 times 12 is 24.

20 + 24dydt = 0

5 Solve for the rate

Take away 20 on both sides.

24dydt = −20

To remove times 24, we divide by 24.

dydt = −2024

A calculator: about −0.83 ft per second.

≈ −0.83 ft/s

The minus means y gets smaller.

The top slides down at 0.83 ft per second.

The top slides down at 0.83 ft/s.

6 Check

Let's check: a tiny moment later, 0.001 seconds.

x is 5.002.

Then y, from Pythagoras, is about 11.99917.

x = 5.002: y = √(13² − 5.002²) ≈ 11.99917

y changed by −0.00083 in 0.001 seconds.

That is about −0.83 ft per second.

Right!

change ÷ 0.001 ≈ −0.83

7 Pick the choice

Our answer is The top slides down at 0.83 ft/s..

That is choice A.

A) 0.83

8 Why not the others

B is 2.

Wrong: the top does not move at the bottom's speed.

B) 2 ✗ the top does not move at the bottom's speed

C is 4.8.

Wrong: used 12/5: it is 5/12.

C) 4.8 ✗ used 12/5: it is 5/12

D is 0.77.

Wrong: used 5/13: the other side is 12, not 13.

D) 0.77 ✗ used 5/13: the other side is 12, not 13

Let's look back.

First: draw and name.

Then: write the equation.

Then: take the derivative.

Then: put the numbers in.

Then: solve for the rate.

Last: check.

Answer