How Do You Solve Challenging Calculus Homework Problems?

[katrina007]

Hi,

I needed some help with my Calculus homework. The teacher gave our class some problems which i have no idea how to start. If someone can tell me how to start the questions or how to approach at solving it... it'll be greatly appreciated. Though I gave my some thoughts. let me know if I am correct/wrong...


Question One
Video tape is unwounded from a spool at a constant rate of 10'' per second. If the tape is 0.02'' thick, how fast is the radius of the spool decreasing when the radius is 2'' ?

** The is a related rate problem so I tried solving or at least started out by first figuring out what formula to use and I think its the sphere volume formula --- V=(4/3)(pie)(r^2). The rate is given 10'' per second and the radius is given too - 2''. I tried plugging in the numbers and taking 1st derivative but that don't seem right or is that how it should be started? Plz help me with this one.

Question Two
Show that the general cubic curve Y = ax^3 + bx^2 + cx + d has one inflection point and three possible shapes depending on whether b^2 > 3ac, b^2 = 3ac or b^2 < 3ac.

** What I did for this question was that I took the 2derivative and got y'' = 6x + 2 (assuming the letters, a-b-c-d are all one coefficients). Then I set the 2nd derivative equal to zero and solved for the possible points of inflections. I then tested (-1) and (1) points from the intervals ( neg. infinity to -1/3) and (-1/3 to pos. infinity) and saw that there is a sign change from negative to positive so therefore there exisits one point of inflection which is x = -1/3.

** I don't know how to do the 2nd part. Not sure where to start and how to show the shapes.

Question Three
If a ray of light emanating from a point A above a horizontal mirror is reflected to a point B above the mirror, show that under the assumption that the light travels by the path of least time, and hence of shortest distance, the angle of incidence is equal to the angle of reflection by using calculus to minimize a function giving the distance. There is a more elegant, purely geometric demonstration for this result.

** There are no numbers here so I don't know what to do. The question itself is confusing to me. Again, please help me with this one too.

Any help to either of these questions will be great.
thanks in advance

 

[Andrew Mason]

Hi,

I needed some help with my Calculus homework. The teacher gave our class some problems which i have no idea how to start. If someone can tell me how to start the questions or how to approach at solving it... it'll be greatly appreciated. Though I gave my some thoughts. let me know if I am correct/wrong...


Question One
Video tape is unwounded from a spool at a constant rate of 10'' per second. If the tape is 0.02'' thick, how fast is the radius of the spool decreasing when the radius is 2'' ?

Question Two
Show that the general cubic curve Y = ax^3 + bx + cx + d has one inflection point and three possible shapes depending on whether b^2 > 3ac, b^2 = 3ac or b^2 < 3ac.

Question Three
If a ray of light emanating from a point A above a horizontal mirror is reflected to a point B above the mirror, show that under the assumption that the light travels by the path of least time, and hence of shortest distance, the angle of incidence is equal to the angle of reflection by using calculus to minimize a function giving the distance. There is a more elegant, purely geometric demonstration for this result.

Any help to either of these questions will be great.
thanks in advance

You haven't given us any of your thoughts. You have to do that first.

AM

 

[katrina007]

Hi,

Sorry I forgot to post those. I editted my post and have given my thoughts. Plz let me know if I've started out right or not. thanks

 

[rocomath]

q1 - Video tape is unwounded from a spool at a constant rate of 10'' per second. If the tape is 0.02'' thick, how fast is the radius of the spool decreasing when the radius is 2''.

if I'm not wrong, i don't think a sphere would be a good model for this related rate problem.

http://img222.imageshack.us/img222/1262/spoolmn3.jpg

q2 & q3 - still thinking ...

btw, is this suppose to be Y = ax^3 + bx + cx + d ---> Y = ax^3 + bx^2 + cx + d?

q3 - so we have point A above a "horizontal-line" where it is emanating a ray of light to point B "above" the mirror. are you able to picture it? so if we're going to be thinking in terms of angles and such, i think we're going to need to include trigonometry in this function.

 

[katrina007]

q1 - Video tape is unwounded from a spool at a constant rate of 10'' per second. If the tape is 0.02'' thick, how fast is the radius of the spool decreasing when the radius is 2''.

if I'm not wrong, i don't think a sphere would be a good model for this related rate problem.

http://img222.imageshack.us/img222/1262/spoolmn3.jpg

q2 & q3 - still thinking ...

btw, is this suppose to be Y = ax^3 + bx + cx + d ---> Y = ax^3 + bx^2 + cx + d?

q3 - so we have point A above a "horizontal-line" where it is emanating a ray of light to point B "above" the mirror. are you able to picture it? so if we're going to be thinking in terms of angles and such, i think we're going to need to include trigonometry in this function.

Hey,

yeah sorry its suppose to by Y=ax^3 + bx^2 + cx + d
I editted the first post now. thanks for letting me know.

And now that i see the picture, I think yea sphere isn't good formula. maybe a cylinder V=(pie)(r^2)(h) ? The thickness can be plugged in for the height, i suppose? and r is already given. So I think I can differeniante or do i need something else?

 

[rocomath]

q1 - yes, that is correct. so what did your answer come out to?

 

[katrina007]

Ok so using the volume formula of a cylinder, I found the rate of the radius decreasing as dr/dt = 39.78

V = (pie)(r^2)(h)
dv/dt' = (pie)(2r)(dr/dt)(dh/dt)

**dv/dt is 10'' per second
** dh/dt is 0.02'' thickness

Please let me know if this is how this problem is supposed to be done or did I do something wrong? Also please help with other problems. Thanks

 

[rocomath]

Video tape is unwounded from a spool at a constant rate of 10'' per second. If the tape is 0.02'' thick, how fast is the radius of the spool decreasing when the radius is 2''

you don't need dh/dt, work it out again and see if you get the right answer.