Differentiation

[paypalpartner]

Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation. p + x^2 = 144 where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded changing when x=9, p=63, and the price/tire is increasing at the rate of $2/week?

 

[soroban]

Hello, paypalpartner !

Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price
by the equation: \(\displaystyle \,p\,+\,x^2\:=\:144\)
where \(\displaystyle p\) is measured in dollars and \(\displaystyle x\) is measured in units of a thousand.

How fast is the quantity demanded changing when \(\displaystyle x\,=\,9,\;p\,=\,63\),
and the price/tire is increasing at the rate of $2/week?

This is a simple Related Rates problem . . . Are you new to them?


We have: \(\displaystyle \,p\,+\,x^2\;=\;144\)

Differentiate with respect to time: \(\displaystyle \,\frac{dp}{dt} \,+ \,2x\left( \frac{dx}{dt} \right) \;=\;0\)

Then we have:\(\displaystyle \,\frac{dx}{dt}\;=\;-\frac{1}{2x}\left(\frac{dp}{dt}\right)\)

We are given: \(\displaystyle \,x\,=\,9,\;p\,=\,63,\;\frac{dp}{dt}\,=\,2\)

Therefore: \(\displaystyle \,\frac{dx}{dt}\;=\;-\frac{1}{2\cdot9}(63)\;=\;-\frac{7}{2}\)


In English: at that particular time, the demand is decreasing at 3500 units per week.

 

[paypalpartner]

yes very new to this thank you very much

 

[Loibanguti]

There is a problem with the final answer:
It has to be -1/9 and not -7/2

 

[Deleted member 4993]

There is a problem with the final answer. It has to be -1/9 and not -7/2

Why do you think so?

Did you work it out?​

Please share your work.

 

[Dr.Peterson]

There is a problem with the final answer:
It has to be -1/9 and not -7/2

You're right; Soroban at one point used the wrong number. Can you identify where in his work this error occurred?

 

[Cubist]

Can you identify where in his work this error occurred?

This question was (I think) directed at OP, @paypalpartner , in case there's any doubt.

It took me several minutes to see the mistake. If I had repeated the calculation myself, from the start, I think I would have found it more quickly! Sometimes it's hard to spot where a problem is ? Well done @Loibanguti and @Dr.Peterson !

 

[Deleted member 4993]

This question was (I think) directed at OP, @paypalpartner ,

I doubt that - the original post appeared about 15 years ago!

Loibanguti has posted 2 problems - without showing a line of work! Thus I wanted to egg him/her on to post some work!

 

[Cubist]

I doubt that - the original post appeared about 15 years ago!

Loibanguti has posted 2 problems - without showing a line of work! Thus I wanted to egg him/her on to post some work!

Ok, it's time for me to stop posting for the day! My observational ? skills have become too weak, it's time for some TV.