# A question on Related rates.

#### Mathelogician

Hi,

The following problem says the rate of change of x with respect to t is 3 meters per second.
Now why should we interpret the red sentence as a derivadive(dx/dt)?And why (dy/dt) for the blue one?
Why not (Delta x/Delta t) or (Delta y/Delta t)?

For 10-meter ladder is leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?
Thanks.

#### Mathelogician

Another example is a water tank filling by water in a specific rate of change....

#### Ackbeet

MHF Hall of Honor
The capital Deltas correspond to average rates of change, whereas the problem is asking about an instantaneous rate of change. They are very different concepts. Do you follow me?

#### Mathelogician

Thanks, but i know that!
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My question is why the red and the blue line are interpreted as derivative?
For Red line, If we use capital deltas, we know that the rate of displacement of x with respect to t is 3, which can conform to the given information.
Also the given information can be interpreted as derivative of a linear function of t with coefficient 3; That is x'(t).

And also the form of the blue question which is not asked us to find the rate of change of y with respect to t at a specific time (for example t1) [which then means derivative of y(t) with respect to t at the point t1] and it could be interpreted like the red one as delta(y)/delta(t).

#### Ackbeet

MHF Hall of Honor
That doesn't work, because with the capital deltas, you're looking at the rate of change over an interval of time. The way the problem is worded, you're interested in the rate of change at one point in time. The key grammar there is the phrase, "is sliding away". That present tense indicates instantaneous. The same goes for the text in blue. It's in the present tense, and it mentions nothing about an interval of time. Hence, everything in sight is instantaneous.

#### Mathelogician

Thanks a lot.

That doesn't work, because with the capital deltas, you're looking at the rate of change over an interval of time.
I'm agree untill here.

The way the problem is worded, you're interested in the rate of change at one point in time.
Sorry but which point in time do you mean?!
The problem says at r=3 no t=3 (if was t, your claim was true!)

The key grammar there is the phrase, "is sliding away". That present tense indicates instantaneous. The same goes for the text in blue. It's in the present tense, and it mentions nothing about an interval of time. Hence, everything in sight is instantaneous.
Why does present tense mention to instantaneous rate of change?!

#### Ackbeet

MHF Hall of Honor
The point in time in which we are interested is the point (the only point) "when the base of the ladder is 6 meters from the wall." That will only happen once, I think you will agree.

The present tense always corresponds to instantaneous rate of change in problems like this, because the present tense only ever refers to one mathematical point in time. When I use the present tense, I want to know what is happening right NOW. I am guessing that your native language is Persian, correct? Surely Persian has a present tense. Does it only refer to one point in time?

#### Mathelogician

Thanks again.
The point in time in which we are interested is the point (the only point) "when the base of the ladder is 6 meters from the wall." That will only happen once, I think you will agree.
Yes that's right!

The present tense always corresponds to instantaneous rate of change in problems like this, because the present tense only ever refers to one mathematical point in time. When I use the present tense, I want to know what is happening right NOW.
Sorry but your assertion is present continuous(Doing NOW) and present tense is the present continuous with a future meaning (Source: English grammar in use - Cambridge University). For example the phrase "Jack is playing tennis on friday evening" is a present tense statement.

I am guessing that your native language is Persian, correct? Surely Persian has a present tense. Does it only refer to one point in time?
That's right!
Present tense in persian is like that in english
[Jack is playing tennis on friday evening= (جک جمعه بعد از ظهر تنیس بازی می کند (خواهد کرد]
And according to my last response, it refers to one point in time, but not RIGHT NOW.
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Now what should we do?!

#### Ackbeet

MHF Hall of Honor
Ok, if you like, I can modify the statement to read, "The present continuous tense always corresponds to instantaneous rate of change in problems like this, because the present continuous tense only ever refers to one mathematical point in time. When I use the present continuous tense, I want to know what is happening right NOW." Like some native English speakers, I've studied the grammar extensively, then I forget all the names for things but I still use them correctly.

Back to the problem: you're going to need to assign some variable names to things so that you can relate the rates. How about the following variables: the distance the bottom of the ladder is away from the wall, and the distance the top of the ladder is from the ground. What do you want to call them?

#### Mathelogician

Ok, if you like, I can modify the statement to read, "The present continuous tense always corresponds to instantaneous rate of change in problems like this, because the present continuous tense only ever refers to one mathematical point in time. When I use the present continuous tense, I want to know what is happening right NOW." Like some native English speakers, I've studied the grammar extensively, then I forget all the names for things but I still use them correctly.
Well... Didn't mean to bather!

Back to the problem: you're going to need to assign some variable names to things so that you can relate the rates. How about the following variables: the distance the bottom of the ladder is away from the wall, and the distance the top of the ladder is from the ground. What do you want to call them?
Obviously, they are the distances and can be named as x and y respectively!
Now what's the connection between this question and the interpretation of the phrase i mentioned first.