VCE Specialist 3/4 Question Thread!

[Eugenet17]

Exact same approach as I used, but solve for x instead of y. Then, you'll need to do , where is more positive in the x-axis, and is more negative in the x-axis.

Think of it like flipping your axis along the line y=x, so that the y-axis becomes the x-axis and vice-versa. Then, you should hopefully see it behaving very similarly to how you know integrals behave.

Also note: if you tried to rotate the circle in your question around the y axis, you'll get a sphere, not a torus.

Awesome, thanks for the clarification!

[Mieow]

Can someone please help me with this:

A population of plants is decreasing because of the effects of a virus. If the population is decreasing according to the differential equation , where t is the time in days, find the time taken for the population to decrease to 25% of its original number

[keltingmeith]

First thing's first: we need to solve the differential equation.



Normally we'd have to split it up here, BUT we know that P >=0, so we can take the positive of this modulus, making our work a lot easier:



We see that is a constant. What we can do here is make this equal to some other constant - preferably, one that's easier to work with. So, let's say

This means we now have the equation . At t=0, P=100%, so let's call this 100 just to make things easier on ourselves. Using this, you should get A = 100, and now we can find how many days it takes for the population to fall to 25%:

[lzxnl]

Can someone please help me with this:

When t = 0, P = P₀. When t = t₀, P = P₀/4. We want to find t₀
Integrating both sides, we get


You don't actually need to solve for an integration constant here xD
This method of solving differential equations by definite integral is mentioned in the Essentials textbook somewhere. It's a bit quicker IMO

[alchemy]

Can someone please help me with this:

Solution attached. Just beaten though...

[keltingmeith]

When t = 0, P = P₀. When t = t₀, P = P₀/4. We want to find t₀
Integrating both sides, we get


You don't actually need to solve for an integration constant here xD
This method of solving differential equations by definite integral is mentioned in the Essentials textbook somewhere. It's a bit quicker IMO

Now that's pretty cool - never seen this applied for modelling questions, I only ever used it for kinematics stuff, hahah. Definitely a lot quicker than my method.

[keltingmeith]

Solution attached. Just beaten though...

Note: while you're right, I would make sure to clarify that P> 0 when you don't include the modulus signs. Just so that the examiner knows that you know why it's okay to omit them in this situation.

[alchemy]

Note: while you're right, I would make sure to clarify that P> 0 when you don't include the modulus signs. Just so that the examiner knows that you know why it's okay to omit them in this situation.

Oh yes, I keep forgetting. I've made this mistake before damn.

[Mieow]

Thanks for the help guys!

Solution attached. Just beaten though...

How did you know that at t=0, P=50?

[keltingmeith]

This is a value plucked out of thin air, which alchemy legitimised later by writing the "assuming original population is 50". For this question, where we're interesting in a percentage not a total amount, it doesn't matter what you might assume the original population to be, as long as you use percentages correctly.

[Bestie]

hey conic,
thank you for your explaination.. but isn't f(b) = f(a) in that diagram? in both option A and B according to the diagrams?

[Mieow]

Oh ok thank you!

So for this question:

The population of a termite colony is increasing at 10% of its current population per month. Find the number of months required for the population to double in size

How would I start this off? Because I let but it didn't give me the right answer 

[alchemy]

Oh ok thank you!

So for this question:How would I start this off? Because I let but it didn't give me the right answer 

Since we want t in terms of P, dt/dP = 10/11P
Find the integral of that and just use the same method as in the last question you asked.

[keltingmeith]

I'm assuming that you've gotten a very large answer, because you've said that the population increases at a rate of 110% of its current population. Try again with

[lzxnl]

This is a value plucked out of thin air, which alchemy legitimised later by writing the "assuming original population is 50". For this question, where we're interesting in a percentage not a total amount, it doesn't matter what you might assume the original population to be, as long as you use percentages correctly.

This is why, in my integral, my bounds were in terms of P₀

Oh ok thank you!

So for this question:How would I start this off? Because I let but it didn't give me the right answer 

INCREASE is 10% of its current population. Also increase is dP/dt