Minimum cost problem

[pinkcalculator]

I have no idea where to start. I know that you need to identify the equation, but I I'm not sure which is which.
The cost C (in dollars) to produce x tons of an alloy used for engine blocks is modeled by the function
C=2.97x^2 -603x + 62,836
0 <x<110 (x is greater than or equal to 0, and less than or equal to 110.)

Find the minimum cost and the corresponding number of tons.

I know to find the vertex, you use the formula (-b/2a, f(-b/2a), and I'm pretty sure a=2.97, b=-603, and I complete the square to finish? I'm not really sure how the x is greater than or equal to zero and less than or equal to 110 applies.


Any help at all would be greatly appreciated, I'm not even sure how to set the problem up.
Thanks.

 

[Denis]

> ...and I'm pretty sure a=2.97, b=-603,

Why are you not absolutely sure?

> I'm not really sure how the x is greater than or equal to zero and less than or equal to 110 applies.

Where do you get "or equal" ?

 

[pinkcalculator]

I'm not so sure about a=2.97 or b=-603 because I'm not sure that that's the equation...If that makes sense. I know that for a minimum value problem, you find the vertex, and I've assumed that it's for the first equation. I'm not positive though, because the x equation is confusing me. I do not know why it's there.

The 'or equal to' is part of the x equation, I just couldn't figure out how to get the line under the greater/lesser than sign.

This is the work I've got done. Can you maybe tell me if I'm at least in the right direction?
2.97x^2 -603x+62,836
x is greater than or equal to 0, and less than or equal to 110.

vertex (603/5.94, C(603/5.94)
C(603/5.94)= 2.97 (603/5.94)^2 -603(603/5.94) +62,836
30606.81-61213.63+62836
C=32,229.18
If (603/5.94)=101.51 tons, does that explain the x portion of the problem?

 

[mmm4444bot]

… the x equation is confusing me. I do not know why it's there …



Hello Pink Calculator:

That is not an equation; it's a compound inequality.

0 <= x <= 110

This is simply a statement of the domain of the cost function. (Have you studied functions? A function's domain is the set of values that are acceptable inputs for the independent variable.)

The reason that the exercise includes the domain is at least two-fold; (1) for thoroughness, and (2) to help students get used to the way things are presented in the real world.

The domain tells us that the given cost function only applies for calculating the cost (in dollars) when producing from 0 tons through 110 tons of alloy. If somebody wanted to calculate the cost of producing more than 110 tons of alloy, then the given cost function does not apply; a different cost function would need to be given or derived.

In this exercise, the stated domain is not used in the steps to determine the requested values; it's simply extra information.

… vertex (603/5.94, C(603/5.94)
C(603/5.94)= 2.97 (603/5.94)^2 -603(603/5.94) +62,836
30606.81-61213.63+62836
C=32,229.18
If (603/5.94)=101.51 tons, does that explain the x portion of the problem?

Yes.

However, since the coefficient a = 2.97 is given in decimal form, there is no reason to work with a ratio of a Whole number over a Decimal number (i.e., 603/5.94).

In other words, after we determine that the x-coordinate of the vertex is 603/5.94, we immediately do the division, and simply work with decimals for the remainder of the exercise.

-b/(2a) = 603/5.94 = 101.52

(Note that you improperly rounded to two decimal places.)

C(101.52) = 2.97(101.52)^2 - 603(101.52) + 62836

C(101.52) = 32229.18



Otherwise, your answers are correct.

Most instructors like to see answers to word problems reported using complete sentences. Something like the following.

"The minimum cost of alloy production is $32,229.18, and this cost corresponds to a production amount of 101.52 tons."

Cheers,

~ Mark

 

[pinkcalculator]

Thanks Mark!
"The domain tells us that the given cost function only applies for calculating the cost (in dollars) when producing from 0 tons through 110 tons of alloy. If somebody wanted to calculate the cost of producing more than 110 tons of alloy, then the given cost function does not apply; a different cost function would need to be given or derived."
This information was very helpful!
I appreciate it.