3U Maths Question Thread

[Shadowxo]

Hello,

I was wondering if someone could please help me with the following question:

Sand falling from a funnel forms a conical pile such that the height of the pile is one and a half times the radius.
a) Show that the volume of the pile is given by V=(pi)r^3/2.
b) If the sand is falling at the rate of (pi)/10m^3 per minute, find the rate at which the height is increasing when the pile of sand is 3m.


I have answered (a) and got the right answer, but I do not get the right answer in (b). The answer of (b) should be 1/40.

Thankyou so much

So for b, it tells you the rate it's falling in m³ per min, or dV/dt. You have to find the rate at which the height is increasing, or dh/dt.
dh/dt = dh/dV * dV/dt.
You have V in terms of r, you can convert it to V in terms of h by substituting in r = 2h/3. V = π(2h/3)³/2 = 4πh³/27
dV/dh = 4πh²/9
dV/dh when h = 3 is 4π  so  dh/dV when h=3 is 1/(4π)
dh/dt = dh/dV * dV/dt = 1/(4π) * π/10 = 1/40 m / min

Hope this helps

[kiwiberry]

Hello,

I was wondering if someone could please help me with the following question:

Sand falling from a funnel forms a conical pile such that the height of the pile is one and a half times the radius.
a) Show that the volume of the pile is given by V=(pi)r^3/2.
b) If the sand is falling at the rate of (pi)/10m^3 per minute, find the rate at which the height is increasing when the pile of sand is 3m.


I have answered (a) and got the right answer, but I do not get the right answer in (b). The answer of (b) should be 1/40.

Thankyou so much

Edit: I got beaten ahah

[yasmineturner]

So for b, it tells you the rate it's falling in m³ per min, or dV/dt. You have to find the rate at which the height is increasing, or dh/dt.
dh/dt = dh/dV * dV/dt.
You have V in terms of r, you can convert it to V in terms of h by substituting in r = 2h/3. V = π(2h/3)³/2 = 4πh³/27
dV/dh = 4πh²/9
dV/dh when h = 3 is 4π  so  dh/dV when h=3 is 1/(4π)
dh/dt = dh/dV * dV/dt = 1/(4π) * π/10 = 1/40 m / min

Hope this helps

Thankyou

[Shadowxo]

Yours is so much neater kiwi (didn't want to have to format all my working)

Also note:
Instead of subbing in r = 2/3h you can find dV/dr instead of dV/dh, then find dV/dh as it equals dV/dr * dr/dh, which is dV/dr * 2/3

[RuiAce]

Yours is so much neater kiwi (didn't want to have to format all my working)

Tis why people learn LaTeX. It does the formatting for you.

[queenie09]

Hey, can you please help me with the following questions:
1. Five-letter words are formed without repetition from the letters of PHYSICAL.
a) How many contain the letter Y?
b)  How many have the two vowels occurring next to one another?
For this one is it just 2!*4! 
2. (a) How many seven-letter words can be formed without repetition from the letters of the word INCLUDE? (don't need help with a) just for background information 
(e) How many have the C immediately following the D?
(f) How many have the letters N and D separated by exactly two letters?
(g) How many have the letters N and D separated by more than two letters?

Thank you in advance!! 

[RuiAce]

Hey, can you please help me with the following questions:
1. Five-letter words are formed without repetition from the letters of PHYSICAL.
a) How many contain the letter Y?
b)  How many have the two vowels occurring next to one another?
For this one is it just 2!*4! 
2. (a) How many seven-letter words can be formed without repetition from the letters of the word INCLUDE? (don't need help with a) just for background information 
(e) How many have the C immediately following the D?
(f) How many have the letters N and D separated by exactly two letters?
(g) How many have the letters N and D separated by more than two letters?

Thank you in advance!! 

In these questions, note that the remaining letters must be chosen out of the bundle of 8 as well, before the arranging.

___________________________



___________________________
___________________________

___________________________



g) is just the more tedious version of f)

Now repeat this process for N and D being together, OR N and D being one letter apart.

Then, if you add these two with the answer for f), you get the number of words formed with N and D separated by 0, 1 or 2 letters.

Finally, you want the opposite (complement) of that. So just do the total outcomes (answer in part a)) minus this number.

[queenie09]

___________________________

Hey the answer for this one is 960

[kiwiberry]

___________________________

Hey the answer for this one is 960

2! x 6C3 x 4! = 960

Hey, I may be wrong but isn't 2(f) 2! x 4 x 5! because all of the leftover letters (ICLUE) need to be used?

[jamonwindeyer]

Hey the answer for this one is 960

I actually popped that expression into Wolfram Alpha and got something different, the grouping was off - Just mentioning that in case it happened to you, because it is 960

[jamonwindeyer]

Hey, I may be wrong but isn't 2(f) 2! x 4 x 5! because all of the leftover letters (ICLUE) need to be used?

I'd agree with that interpretation! Perhaps Rui accidentally reverted to the 5 letter word from Question 1

[RuiAce]

Yep. Ironically enough I went away from the correct answer. Let me fix that.

[legorgo18]

Hello, can someone point me in the right direction of part b ii) of this projectile q

part a was just find the 6 equations
x'= v x= vt
y'= -gt y= h-1/2gt^2

b) A canister containing a life raft is dropped from a plane to a stranded sailor. The plane is travelling at a constant velocity of 216 km/h, at a height of 120 metres above sea level along a path that passes above the sailor.

i) How long will the canister take to hit the water?
I got t = 2 root 6

ii) A current is causing the sailor to drift at a speed of 2.6 km/h in the same direction as the plane is travelling. The canister is dropped from the plane when the horizontal distance from the plane to the sailor is D metres. What values can D take if the canister is allowed to land at most 50 metres from the stranded sailor?

Any help is appreciated

[jamonwindeyer]

Hello, can someone point me in the right direction of part b ii) of this projectile q

part a was just find the 6 equations
x'= v x= vt
y'= -gt y= h-1/2gt^2

b) A canister containing a life raft is dropped from a plane to a stranded sailor. The plane is travelling at a constant velocity of 216 km/h, at a height of 120 metres above sea level along a path that passes above the sailor.

i) How long will the canister take to hit the water?
I got t = 2 root 6

ii) A current is causing the sailor to drift at a speed of 2.6 km/h in the same direction as the plane is travelling. The canister is dropped from the plane when the horizontal distance from the plane to the sailor is D metres. What values can D take if the canister is allowed to land at most 50 metres from the stranded sailor?

Any help is appreciated

Hey! So based on your answer for the first question I'll assume \(g=10\). This question is all about the horizontal position of the raft at \(t=2\sqrt{6}\), so using the horizontal equation for the raft. we can figure that out:



Notice the conversion to metres per second for the speed of the raft/plane.

So let's interpret that correctly, this means that the raft lands this many metres in front of the point of release.

This is tough to explain, so let me add a diagram here (not to scale AT ALL):



Basically, we need to do two things. First, notice the sailor has moved a distance in the time it takes for the raft to reach the sea. We can calculate that distance:



So there will be two answers. ONE corresponds to the plane releasing the raft really early, so it comes and lands 50 metres to the left (in the diagram) of the NEW position. The other corresponds to the plane releasing the raft late, so it comes and lands 50 metres to the RIGHT of the NEW position.

We need to work backwards. Your rough working should be (this will pretty much give you the answer, try with the diagram first!):

Process

1. Figure out where the raft should be released to correspond to the leftmost position, 50 metres from the raft on the left. Hint, if it travels \(120\sqrt{6}\) metres horizontally during the fall, it will be that much further to the left!).
2. Adjust for the movement of the sailor - How far is that from the original position? Hint: Subtract the distance I gave above.
3. Repeat fort he right most position with small adjustments (a labelled diagram REALLY helps

Let us know if you want the full working/answers, happy to provide, but this is a general guideline and hopefully some direction

[Shadowxo]

Hello, can someone point me in the right direction of part b ii) of this projectile q

part a was just find the 6 equations
x'= v x= vt
y'= -gt y= h-1/2gt^2

b) A canister containing a life raft is dropped from a plane to a stranded sailor. The plane is travelling at a constant velocity of 216 km/h, at a height of 120 metres above sea level along a path that passes above the sailor.

i) How long will the canister take to hit the water?
I got t = 2 root 6

ii) A current is causing the sailor to drift at a speed of 2.6 km/h in the same direction as the plane is travelling. The canister is dropped from the plane when the horizontal distance from the plane to the sailor is D metres. What values can D take if the canister is allowed to land at most 50 metres from the stranded sailor?

Any help is appreciated

What I did is make an equation

D + (distance travelled by plane) - (distance travelled by boat) = + or - 50 , with negative D or 50 referring to that many metres left of the sailor.
So if Jamon's working is right
D + 120√6 - \(\frac{13√6}{9}\) = + or - 50, then solve for possible D values (and change them into positive as they're the distance left of the sailor)