Mathematics Question Thread

[jakesilove]

How would I do question 14?

Hey! Recall that



Where x is displacement. So, in this case,





Imposing our initial conditions,

[Youssk]

Hey, could someone please help me with this question.

A cylinderical can is to have a volume of 20πm^3. The material for the circular top and bottom costs $10 per m^2 and the material for the curve surface costs $8 per m^2.

i) given that there are no overlapping edges and that the surface area of a cylinder is given by SA=2πr^2 + 2πrh, show the cost of the cylinder is given by C= 20πr^2 + 320π/(divided by) r.

ii) find the minimum cost of the material for the can.

Thankyou! 

[jakesilove]

Hey, could someone please help me with this question.

A cylinderical can is to have a volume of 20πm^3. The material for the circular top and bottom costs $10 per m^2 and the material for the curve surface costs $8 per m^2.

i) given that there are no overlapping edges and that the surface area of a cylinder is given by SA=2πr^2 + 2πrh, show the cost of the cylinder is given by C= 20πr^2 + 320π/(divided by) r.

ii) find the minimum cost of the material for the can.

Thankyou! 

Hey! So, we know that the circular parts of the can cost $10 per square meter, so



given that there is a circular top, and a circular bottom. We also know that the cost of the curved surface is $8 per square meter, so



So, the total cost will be



We need to get rid of the h somehow, so we impose our initial conditions. The volume of a cylinder is



We have been given volume, so




Subbing this into our cost relationship,



Great! Now, we want to minimise this cost. So, we differentiate to find stationary points



Multiplying through by r squared and dividing by pi




Since there is only one stationary point, it's fair to assume it is a minimum. However, I would recommend testing points to either side using the first derivative, and finding that 1.9 yields a negative value, whilst 2.1 yields a positive value (showing that concavity is positive, and thus the point is a minimum. This is way easier than finding the second derivative in this case!)

Now, we plug r=2 back into our cost equation to get the total cost;

[Thebarman]

Thanks for the help with the last question!

How would I find the equation of a parabola with a focus (3,2) and directrix y=-4?

[jakesilove]

Thanks for the help with the last question!

How would I find the equation of a parabola with a focus (3,2) and directrix y=-4?

Hey! Firstly, we can easily find the vertex of the parabola. It will be halfway between the directrix and the focus. So, vertex is (3, -1). Now, the equation of a parabola is



Where the vertex has coordinates (b,c). So,



The distance between the vertex and the directrix is a. So, a=3

[Fahim486]

Hey, I'm having trouble with part i. and ii. so could someone pls show me how to do it
Thanks!

[jakesilove]

Hey, I'm having trouble with part i. and ii. so could someone pls show me how to do it
Thanks!

Hey,

It's a bit difficult for me to draw you a probability tree diagram. Basically, you will first have two branches stemming from a central point. These two branches represent Monday; one branch (labelled M for 'miss') has a 0.3 chance of occurring, whilst the other branch (labelled C for 'catch') has a 0.7 chance of occurring. From each of these branches, another two branches will be constructed. These four branches represented Tuesday; against, one of each branches is labelled M (probability 0.2) and the other is labelled C (probability 0..

The probability of missing the train on ONE of the days is going to be 1 minus the probability of missing the train on NO days, minus the probability of missing the train on BOTH days (note that it is not AT LEAST one, it is ONLY one). So,



We can calculate the above probabilities by just multiplying the two days together

[Fahim486]

Hey,

It's a bit difficult for me to draw you a probability tree diagram. Basically, you will first have two branches stemming from a central point. These two branches represent Monday; one branch (labelled M for 'miss') has a 0.3 chance of occurring, whilst the other branch (labelled C for 'catch') has a 0.7 chance of occurring. From each of these branches, another two branches will be constructed. These four branches represented Tuesday; against, one of each branches is labelled M (probability 0.2) and the other is labelled C (probability 0..

The probability of missing the train on ONE of the days is going to be 1 minus the probability of missing the train on NO days, minus the probability of missing the train on BOTH days (note that it is not AT LEAST one, it is ONLY one). So,



We can calculate the above probabilities by just multiplying the two days together

ah okay now i see where i went wrong. Thanks so much for your help!!!

[Fahim486]

Hi once again, for question ii. I've worked out the coordinates of the stationary point but how would you determine the nature? I'm confused as to whether you differentiate once to determine the nature or you have to double differentiate.
Thanks again!!

[Thebarman]

Thank you!

Hi once again, for question ii. I've worked out the coordinates of the stationary point but how would you determine the nature? I'm confused as to whether you differentiate once to determine the nature or you have to double differentiate.
Thanks again!!

Hey Fahim, I hope you don't mind if I take a look at your question.
To put it simply, we use the first derivative (y') to determine the gradient, as well as to find any stationary points.
The second derivative (y'') is used to determine concavity (is it curved upwards or downwards), and whether the curve has a minimum or maximum value. Essentially, this determines the shape of the curve.

So if you need to determine the nature, you could either create a table of values and look a little bit to the left and a little bit to the right of the stationary points to determine their value, OR you can use the second derivative. The first option is much more time consuming, so we'll opt for the second.

In your case, the stationary points would be (2,0) and (8/3, 4/27). [I think. Forgive me if my math is a little off...]

To determine the nature, you would then find the second derivative (i.e. y'' = -6x + 14), and then sub the two x-coordinates of the stationary points into this equation.
If the answer is greater than 0 (y'' > 0), then the curve is a minimum, as it is concave up (there is a minimum x-value).
If the answer is less than 0 (y'' < 0), then the curve is a maximum, as it is concave down (there is a maximum x-value).

So, when x=2,
y'' = -6(2) + 14 = 2
y'' > 0
Therefore, there is a maximum at (2,0)

Similarly, when x=8/3,
y'' = -6(8/3) + 14 = -2
y'' < 0
Therefore, there is a minimum at (8/3, 4/27)

And that's it! Your answer would simply be the last line in each explanation (i.e. max/min at (x,y)).
Hope this explanation was helpful!

[Fahim486]

Hey Fahim, I hope you don't mind if I take a look at your question.
To put it simply, we use the first derivative (y') to determine the gradient, as well as to find any stationary points.
The second derivative (y'') is used to determine concavity (is it curved upwards or downwards), and whether the curve has a minimum or maximum value. Essentially, this determines the shape of the curve.

So if you need to determine the nature, you could either create a table of values and look a little bit to the left and a little bit to the right of the stationary points to determine their value, OR you can use the second derivative. The first option is much more time consuming, so we'll opt for the second.

In your case, the stationary points would be (2,0) and (8/3, 4/27). [I think. Forgive me if my math is a little off...]

To determine the nature, you would then find the second derivative (i.e. y'' = -6x + 14), and then sub the two x-coordinates of the stationary points into this equation.
If the answer is greater than 0 (y'' > 0), then the curve is a minimum, as it is concave up (there is a minimum x-value).
If the answer is less than 0 (y'' < 0), then the curve is a maximum, as it is concave down (there is a maximum x-value).

So, when x=2,
y'' = -6(2) + 14 = 2
y'' > 0
Therefore, there is a maximum at (2,0)

Similarly, when x=8/3,
y'' = -6(8/3) + 14 = -2
y'' < 0
Therefore, there is a minimum at (8/3, 4/27)

And that's it! Your answer would simply be the last line in each explanation (i.e. max/min at (x,y)).
Hope this explanation was helpful!

That was explained very clearly and I finally understood the question. Thanks so much for your help!!!

[jakesilove]

Thank you!
Hey Fahim, I hope you don't mind if I take a look at your question.
To put it simply, we use the first derivative (y') to determine the gradient, as well as to find any stationary points.
The second derivative (y'') is used to determine concavity (is it curved upwards or downwards), and whether the curve has a minimum or maximum value. Essentially, this determines the shape of the curve.

So if you need to determine the nature, you could either create a table of values and look a little bit to the left and a little bit to the right of the stationary points to determine their value, OR you can use the second derivative. The first option is much more time consuming, so we'll opt for the second.

In your case, the stationary points would be (2,0) and (8/3, 4/27). [I think. Forgive me if my math is a little off...]

To determine the nature, you would then find the second derivative (i.e. y'' = -6x + 14), and then sub the two x-coordinates of the stationary points into this equation.
If the answer is greater than 0 (y'' > 0), then the curve is a minimum, as it is concave up (there is a minimum x-value).
If the answer is less than 0 (y'' < 0), then the curve is a maximum, as it is concave down (there is a maximum x-value).

So, when x=2,
y'' = -6(2) + 14 = 2
y'' > 0
Therefore, there is a maximum at (2,0)

Similarly, when x=8/3,
y'' = -6(8/3) + 14 = -2
y'' < 0
Therefore, there is a minimum at (8/3, 4/27)

And that's it! Your answer would simply be the last line in each explanation (i.e. max/min at (x,y)).
Hope this explanation was helpful!

Absolute legend, cheers for answering the question! Just goes to show how well you know the topic area

[Kekemato_BAP]

Hi, I am stuck on this series-sequences question (circled in pic). Could someone please explain how the answer is 180? Thanks!!

[kiwiberry]

Hi, I am stuck on this series-sequences question (circled in pic). Could someone please explain how the answer is 180? Thanks!!

Hey!
We are given that S₇ = 5T₇ and T₆ + T₇ = 40

Looking at the first equation,
S₇ = 5T₇
7/2(2a + 6d) = 5(a + 6d)
2a - 9d = 0 ---(1)

Looking at the second equation,
T₆ + T₇ = 40
(a + 5d) + (a + 6d) = 40
2a + 11d = 40 ---(2)

Solving (1) and (2) simultaneously, (2)-(1): 20d = 40 so d=2. Subbing this back in we get a=9

So S₁₀ = 10/2(2a + 9d) = 5(18 + 18) = 180

[Kekemato_BAP]

Hey!
We are given that S₇ = 5T₇ and T₆ + T₇ = 40

Looking at the first equation,
S₇ = 5T₇
7/2(2a + 6d) = 5(a + 6d)
2a - 9d = 0 ---(1)

Looking at the second equation,
T₆ + T₇ = 40
(a + 5d) + (a + 6d) = 40
2a + 11d = 40 ---(2)

Solving (1) and (2) simultaneously, (2)-(1): 20d = 40 so d=2. Subbing this back in we get a=9

So S₁₀ = 10/2(2a + 9d) = 5(18 + 18) = 180

Thank you!! That really helps