Question

A 25-ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 0.18 ft/sec, how fast, in ft/sec, is the top of the ladder sliding down the wall, at the instant when the bottom of the ladder is 20 ft from the wall? Answer with 2 decimal places. Type your answer in the space below. If your answer is a number less than 1, place a leading "0" before the decimal point (ex: 0.35).

Answer 1

@TheSmartOne

Answer 2

Classic problem found in elementary calculus. I'd suggest you sketch this situation and label everything, e. g., the length of the ladder, the height off the ground where the ladder touches the wall, the distance of the base from the wall, the rate at which the bottom of the ladder slides away from the wall, etc. Once y ou've done that, we can write equations relating these constants and variables.

Answer 3

Note that if you draw this ladder situation. the wall, the ladder and the distance of the bottom of the ladder from the wall form a TRIANGLE, so that you can put the Pythagorean Theorem to good use.

Answer 4

I got so far as w^2+g^2=25^2

Answer 5

@mathmale

Answer 6

Arianna, I'd surely like to see a SKETCH. I have no way to know (other than guess) what your " w " and your " g " represent.

Answer 7

Hint: 25^2 is good; that's the SQUARE of the length of the ladder.

Answer 8

Better. You've used the Pyth. Thm. to present the relationship between g and w and 25.

Answer 9

Your w^2+g^2=25^2 can be differentiated with respect to t (time). The resulting equation will be one in "related rates."

Answer 10

RIght. From there it confused me.

Answer 11

I assume you're familiar with the power rule for differentiation.

(d/dx) x^2 = 2x.

Are you familiar with the "power and chain rule?"

Answer 12

you have g^2 + w^2 = 25^2 and must differentiate with respect to time, t, to get time rates of change that are related. ("Related rates")

Answer 13

yes. 2ww' + 2gg' = 0

Answer 14

Which rate is given to y ou in this problem, w' or g'? What's the value? + or - ?

Answer 15

0.18 g'

Answer 16

The length of the ladder is constant: 25'
The base of the triangle is increasing at the rate of 0.18 ft/sec (there's no g' here).
What can you say about the rate of change of the height of the triangle?

Answer 17

2(15)w'+2(20)(0.18) = 0 I solved for w and got -0.24. IS that wrong?

Answer 18

Please write that result with the proper units of measurement.
Do YOU think your result is right or wrong? if you have doubts about it, what are your doubts?