Derivatives

[Kyan]

Hello, i need help please

A box without a lid has the shape of a parallel-rectangle pipede.

Its base is a square of side x (expressed in meters) with x> 0.​

The volume of the box is equal to 10 m³​

The base is made using a material which costs 5 € per meter square, while the side faces are built using a material that costs 2 € per square meter. We denote h the height of the box and c the cost of making a box.​

1. Express h depending on x.
2. Show that, for all x> 0, C (x) = 5 (x³ +16)/x
3. We denote by C’ the function derived from C. Show that, for all x> 0,

C '(x) = 10 (x³-8)/x²​

4. Study the variations of the function C then find the dimensions of the box for which the manufacturing cost is minimal.

For the first 1 : i started with c²*h=10 but i don’t know what to do and is this actually correct

 

[Deleted member 4993]

Hello, i need help please

A box without a lid has the shape of a parallel-rectangle pipede.

Its base is a square of side x (expressed in meters) with x> 0.​

The volume of the box is equal to 10 m³​

The base is made using a material which costs 5 € per meter square, while the side faces are built using a material that costs 2 € per square meter. We denote h the height of the box and c the cost of making a box.​

1. Express h depending on x.
2. Show that, for all x> 0, C (x) = 5 (x³ +16)/x
3. We denote by C’ the function derived from C. Show that, for all x> 0,

C '(x) = 10 (x³-8)/x²​

4. Study the variations of the function C then find the dimensions of the box for which the manufacturing cost is minimal.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[Dr.Peterson]

Hello, i need help please

A box without a lid has the shape of a parallel-rectangle pipede.

Its base is a square of side x (expressed in meters) with x> 0.​

The volume of the box is equal to 10 m³​

The base is made using a material which costs 5 € per meter square, while the side faces are built using a material that costs 2 € per square meter. We denote h the height of the box and c the cost of making a box.​

1. Express h depending on x.
2. Show that, for all x> 0, C (x) = 5 (x³ +16)/x
3. We denote by C’ the function derived from C. Show that, for all x> 0,

C '(x) = 10 (x³-8)/x²​

4. Study the variations of the function C then find the dimensions of the box for which the manufacturing cost is minimal.

For the first 1 : i started with c²*h=10 but i don’t know what to do and is this actually correct

Did you mean x²*h=10? Yes, that is a start.

Now solve that for h, and you'll have answered the first question.

Then try writing an expression for C(x), using the expression for h.

By the way, I think "parallel-rectangle pipede" should be "rectangular parallelepiped", or just "right square prism".

 

[Kyan]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

Did you mean x²*h=10? Yes, that is a start.

Now solve that for h, and you'll have answered the first question.

Then try writing an expression for C(x), using the expression for h.

By the way, I think "parallel-rectangle pipede" should be "rectangular parallelepiped", or just "right square prism".

Yes i actually answered the 3 question and the first one is h= 10/x^2
Now i can’t answer the 4th one because i don’t know how to study the variation of this fonction, i just did the variation of the first and second degree functions and no third degree

 

[Dr.Peterson]

Yes i actually answered the 3 question and the first one is h= 10/x^2
Now i can’t answer the 4th one because i don’t know how to study the variation of this fonction, i just did the variation of the first and second degree functions and no third degree

I was curious about the wording, "study the variations". Can you show us what you mean by that in your class?

I would expect it to say, find the derivative and use that to maximize the function. This is not a question about polynomials, and any methods you learned for those, of any degree, are not useful here. But the derivative is.

 

[Kyan]

I was curious about the wording, "study the variations". Can you show us what you mean by that in your class?

I would expect it to say, find the derivative and use that to maximize the function. This is not a question about polynomials, and any methods you learned for those, of any degree, are not useful here. But the derivative is.

Actually we have the derivatives of the C function and its C’, when i say « study the variation » im talking about this

 

[Dr.Peterson]

Actually we have the derivatives of the C function and its C’,

Right, I missed that you already had C'. I was confused about why you said this:

Now i can’t answer the 4th one because i don’t know how to study the variation of this function, i just did the variation of the first and second degree functions and no third degree

You shouldn't need to have learned anything special about the cubic (which is in the numerator of C'). You can easily find when it is zero, and if you are referring to finding when it is positive or negative, that, too, is easy enough. Have you tried factorizing it?

Commonly we learn to analyze a rational function like your C' by factoring it and either examining the signs in intervals separated by zeros of the numerator or denominator, or just thinking about the signs of the factors. Have you not seen rational functions yet? In this respect, it is not very different from polynomials (of any degree).

 

[Kyan]

Right, I missed that you already had C'. I was confused about why you said this:

You shouldn't need to have learned anything special about the cubic (which is in the numerator of C'). You can easily find when it is zero, and if you are referring to finding when it is positive or negative, that, too, is easy enough. Have you tried factorizing it?

Commonly we learn to analyze a rational function like your C' by factoring it and either examining the signs in intervals separated by zeros of the numerator or denominator, or just thinking about the signs of the factors. Have you not seen rational functions yet? In this respect, it is not very different from polynomials (of any degree).

Yes but isnt C’ = 10 (x³-8)/x2 already factorized so the solution is 8 and it’s positive ? I don’t know how this can help me to find the dimensions of the box for which the manufacturing cost is minimal. And isnt C already factorized and C’ too

 

[Dr.Peterson]

Yes but isnt C’ = 10 (x³-8)/x² already factorized so the solution is 8 and it’s positive ? I don’t know how this can help me to find the dimensions of the box for which the manufacturing cost is minimal. And isnt C already factorized and C’ too

No, neither C nor C' is fully factorized as you have shown them; and the "solution" of C' (assuming you mean its zero) is not 8.

C' is zero when x^3 = 8, which means x = 2; and it's undefined (which is another case where its sign can change) when x = 0.

With that information, you can find out where C' changes sign from negative to positive, and so determine where C has a minimum. That's what you're looking for, right?

 

[Kyan]

No, neither C nor C' is fully factorized as you have shown them; and the "solution" of C' (assuming you mean its zero) is not 8.

C' is zero when x^3 = 8, which means x = 2; and it's undefined (which is another case where its sign can change) when x = 0.

With that information, you can find out where C' changes sign from negative to positive, and so determine where C has a minimum. That's what you're looking for, right?

Yes, so the function changes sign only one time? Or twice lile the second degree on -2 and 2 ?

 

[Dr.Peterson]

Only positive values of x are relevant to your problem. What is the derivative at x=2? What is it just to the left, or just to the right? So what is happening there?