Implict differentiation and price

[ver_mathstats]

Homework Statement

When the price of a 400g bag of almonds is p dollars per bag, customers demand x hundred bags of almonds where x^2+4px+p^3=81. At what rate is the demand x changing with respect to time when the price is $4 per bag and decreasing at a rate of 20 cents per month?

Relevant Equations

x^2+4px+p^3=81

For this one I did implicit differentiation. Where I then obtained y'=(-2x-4p)/(4x+3p²).

Once I had this I plugged in my values where p is $4 per bag and x is 20 cents.

I plugged in my values y'= (-2(20)-4(4))/(4(20)+3(4)²) =-7/16.

However when I checked this answer it was incorrect and I am unsure of where I am going wrong.

I also tried to do it with -20 substituted since it is decreasing and y'= (-2(-20)-4(4))/(4(-20)+3(4)²) = -0.75, however this was incorrect as well.

I am unsure of where to go from here. Thank you.

 

[tnich]

Using implicit differentiation is the right idea, but you have some problems in the implementation. First of all, since you are looking for a rate of change with time, you need to differentiate the demand curve with respect to time $t$. You will get terms with $x$, $p$, $\dot x$, and $\dot p$ in them in various combinations. There shouldn't be a $y$ in any of the terms.

Second, you need to use commensurate units. The price $p$ is given in $, and the time rate of change of price is given in cents/month. If the answer is to make any sense, you need to convert the $ to cents for the price, or the cents/month to $/month for the price rate of change. You will also need to figure out what units to state the answer in.

Third, you have incorrectly used 20 cents for the initial demand. Demand is not a price, it is the number of bags of almonds that can be sold at a given price, so it should be in units of bags of almonds, not cents. You can calculate the demand $x$ by substituting the initial value of $p$ into the demand curve equation and solving for $x$.

 

[ver_mathstats]

Using implicit differentiation is the right idea, but you have some problems in the implementation. First of all, since you are looking for a rate of change with time, you need to differentiate the demand curve with respect to time $t$. You will get terms with $x$, $p$, $\dot x$, and $\dot p$ in them in various combinations. There shouldn't be a $y$ in any of the terms.

Second, you need to use commensurate units. The price $p$ is given in $, and the time rate of change of price is given in cents/month. If the answer is to make any sense, you need to convert the $ to cents for the price, or the cents/month to $/month for the price rate of change. You will also need to figure out what units to state the answer in.

Third, you have incorrectly used 20 cents for the initial demand. Demand is not a price, it is the number of bags of almonds that can be sold at a given price, so it should be in units of bags of almonds, not cents. You can calculate the demand $x$ by substituting the initial value of $p$ into the demand curve equation and solving for $x$.

So I realized all of the mistakes I made. Instead of substituting -20, I must do -0.20 I think. I differentiated with respect to t. I solved for x to obtain x=1 and x=-17. I substituted my values into the equation this included dp/dt=-0.2, p=4, x=1 since -17 cannot be used. I got the answer 26/45 or 0.58 rounded to two decimals.

 

[tnich]

So I realized all of the mistakes I made. Instead of substituting -20, I must do -0.20 I think. I differentiated with respect to t. I solved for x to obtain x=1 and x=-17. I substituted my values into the equation this included dp/dt=-0.2, p=4, x=1 since -17 cannot be used. I got the answer 26/45 or 0.58 rounded to two decimals.

That's what I got, too.