# Math Help - Several Calculus word problems, which i cant solve :(

1. ## Several Calculus word problems, which i cant solve :(

The marginal profit made by a manufacturer is the additional profit made when production is increased by a small amount. It is the rate of increase of profit with respect to the volume of production. A manufacturer of plastic pipes has determined that with its present equipment the profit function for the production of pipes is:

P(n) = 67500n + 600n^2 - n^3 - 1200000

In the first seconds after launch, the height of a rocket above the ground is given by the expression:

h(t) = 5t^3 + 30t^2 + 45t + 4

where h is in metres and t is in seconds. How long does the rocket take to reach a speed of 75 m/s

NOTE: Speed is the rate of change of distance with respect to time

A farmer has 140m of chicken wire which she is going to use against an existing boundary fence, to enclose a vegetable garden, which she wants in the shape of a rectangle.

Find the maximum area of her vegetable garden.
(You must justify, using calculus, that your result is the maximum area)

A drink manufacturer proposes a new style of can packaging. Given that the sum of the circumference and the height of the new can is to be 30cm, what is the maximum possible volume of the new can package?

(You must justify, using calculus, that your result is the maximum volume)

A particle moves in a straight line so that its displacement from a point O, at any time t, is

x = *square root* 2t^3 + 4 (x is in metres and t is in seconds)

Find the acceleration when t = 2

(NOTE: Acceleration is the rate of change of velocity with respect to time)

The sum of two numbers is 10. What are the two numbers if the sum of their squares is a minimum?

For the curve, y = ax^2 + bx + c, where a, b and c are constants, it is given that at the points (2, 12) and (-1, 0) the slope of the tangent is 7 and 1 respectively.

Find a, b and c.

2. this is a lot of questions. you're not trying to get us to do your homework are you?
Originally Posted by I Drink and Derive
The marginal profit made by a manufacturer is the additional profit made when production is increased by a small amount. It is the rate of increase of profit with respect to the volume of production. A manufacturer of plastic pipes has determined that with its present equipment the profit function for the production of pipes is:

P(n) = 67500n + 600n^2 - n^3 - 1200000
question is incomplete

In the first seconds after launch, the height of a rocket above the ground is given by the expression:

h(t) = 5t^3 + 30t^2 + 45t + 4

where h is in metres and t is in seconds. How long does the rocket take to reach a speed of 75 m/s

NOTE: Speed is the rate of change of distance with respect to time
hint: rate of change of distance with respect to time is given by the derivative. this gives the rate of change. you are asked to find t such that h'(t) = 75

A farmer has 140m of chicken wire which she is going to use against an existing boundary fence, to enclose a vegetable garden, which she wants in the shape of a rectangle.

Find the maximum area of her vegetable garden.
(You must justify, using calculus, that your result is the maximum area)
hint: start by drawing a diagram (ALWAYS a good idea for optimization problems, when possible, which it definitely is in this case). label the diagram. for instance, label the length x and the width (the side parallel to the existing fence) y. note that

$2x + y = 140$ .............the total length of the wire. this is your constraint equation.

now, you want to maximize your area equation, given by:

$A = xy$

A drink manufacturer proposes a new style of can packaging. Given that the sum of the circumference and the height of the new can is to be 30cm, what is the maximum possible volume of the new can package?

(You must justify, using calculus, that your result is the maximum volume)
hint: let r be the radius of the end of the can and h be the height of the can. as the problem directs,

$2 \pi r + h = 30$ ...........this is your constraint equation.

you want to maximize the volume, given by:

$V = \pi r^2 h$

A particle moves in a straight line so that its displacement from a point O, at any time t, is

x = *square root* 2t^3 + 4 (x is in metres and t is in seconds)

Find the acceleration when t = 2

(NOTE: Acceleration is the rate of change of velocity with respect to time)
hint: differentiate the position equation twice to get the acceleration equation. you want the acceleration when t = 2

[/quote]
The sum of two numbers is 10. What are the two numbers if the sum of their squares is a minimum?[/quote]let the numbers be x and y

the first sentence says,

x + y = 10 ..............this is your constraint equation

$S = x^2 + y^2$

where S is the sum of the squares of the two numbers.

For the curve, y = ax^2 + bx + c, where a, b and c are constants, it is given that at the points (2, 12) and (-1, 0) the slope of the tangent is 7 and 1 respectively.

Find a, b and c.
call y f(x) for clarity.

since (2,12) and (-1,0) are on the curve, we have:

f(2) = 12 and
f(-1) = 0

given the respective tangents, we have

f ' (2) = 7 and
f ' (-1) = 1

you can set up simultaneous equations with the information above to solve for a, b and c

3. Thanks for that, no i'm not trying to get you to do my homework :P... I was just picking stuff from my textbook that i couldn't do (preparing for exams)

4. Originally Posted by I Drink and Derive
Thanks for that, no i'm not trying to get you to do my homework :P... I was just picking stuff from my textbook that i couldn't do (preparing for exams)

ok, no worries. just checking.

be sure to ask questions if you are not getting something

5. One thing, i see you mentioned optimization above.

This is something that has been omitted from our course..

(i suspect they are going to chuck it into the problem solving exam)

would you be able to link me or show me how to optimize?

that would help a great deal.

6. Originally Posted by I Drink and Derive
One thing, i see you mentioned optimization above.

This is something that has been omitted from our course..

(i suspect they are going to chuck it into the problem solving exam)

would you be able to link me or show me how to optimize?

that would help a great deal.
the common thing you should take from the following is that optimization is an exercise in derivatives. you are given information to form a constraint equation and an objective equation. the constraint equation, as its name suggests, is an equation that constrains you. it tells you exactly how the variables of a problem must relate to each other. for example, one might be x + y = 10. this constraints your choice of x and y. you can't pick any ol' values! they must add up to ten! the constraint equation allows you to solve for one variable in terms of the others. so that you can plug them into the objective equation to get it in a single variable.

the objective equation, as its name suggest, tells you what is your objective to optimize. if they say maximize or minimize the area, then the area equation for the object you are considering is your objective equation. once you have the function, you can optimize it. here is where derivatives come in. the (local) maximums or minimums of a function is given by the derivative. these extrema occur when the derivative (slope) is zero. so finding the derivative allows you to see what value of your variable will cause the slope to be zero and hence maximize/minimize your quantity. you can test, if necessary, that something is a max or min using the second derivative test. i will assume you know what that is

Pauls Online Notes : Calculus I - Optimization

http://www.mathhelpforum.com/math-he...-problems.html

http://www.mathhelpforum.com/math-he...-problem3.html

http://www.mathhelpforum.com/math-he...n-problem.html

http://www.mathhelpforum.com/math-he...-problem2.html

http://www.mathhelpforum.com/math-he...n-problem.html

http://www.mathhelpforum.com/math-he...-problems.html

http://www.mathhelpforum.com/math-he...e-problem.html

http://www.mathhelpforum.com/math-he...imization.html

http://www.mathhelpforum.com/math-he...lems-calc.html

http://www.mathhelpforum.com/math-he...-problem6.html

http://www.mathhelpforum.com/math-he...-problem5.html

http://www.mathhelpforum.com/math-he...-problem4.html

http://www.mathhelpforum.com/math-he...imization.html

whew! anyway, take care, and good luck. you can ask more questions if you want. i'll probably be logging off soon, so someone else would have to answer