Question

The cost, in millions of dollars, for a company to manufacture x thousand speed boats is given by the function C(x) = 3x2 - 24x + 144. Find the number of speedboats that must be produced to minimize the cost. To minimize the cost, the company must produce thousand speedboats.

Answer 1

4 thousand speedboats

Explanation:

Vertex Form

The minimum/maximum point on a parabola is just another name for that parabola's vertex. A parabola can be defined in a few different ways, but one is as the curve described by a quadratic function, a function of the form
`y=ax^2+bx+c` where a, b, and c ≠ 0. To see how we can get a vertex out of this, we can start with the simpler function
`y=ax^2`. Here, the vertex is simply the origin, (0, 0). If we shift the graph horizontally by h units, replacing x with (x - h), we get the function
`y = a(x-h)^2` and the vertex (h, 0), and if we shift it vertically by k units, we get the equation
`y-k=a(x-h)^2` and the vertex (h, k). We can, of course, add k to either side to obtain the function
`y=a(x-h)^2+k`, also known as the general vertex form of a quadratic function.

The Problem: Completing the Square

This problem asks us to find a value for x which would minimize the C(x) in the function
`C(x)=3x^2-24x+144`. This essentially boils down to getting C(x) in vertex form and finding the x coordinate of the vertex from there. To do this, we can utilize an algebraic technique called completing the square to transform the expression on the right side into the form we want. Our task then is to somehow manipulate
`3x^2-24x+144` so that it resembles the form
`a(x-h)^2+k`, where a, h, and k are constants, and (h, k) is the vertex of the parabola.

The first thing we can do with our expression is pull out a 3 from all three terms:

`3x^2-24x+144\rightarrow3(x^2-8x+48)`

What we'd like now is to somehow turn that expression in the parentheses into something resembling
`(x-h)^2`. To do this, we can recall that

`(x-y)^2=x^2-2xy+y^2`.

If we rewrite
`(x^2-8x+48)` as
`(x^2-2\cdot4\cdot x+48)`, we can see that this almost resembles
`(x-4)^2=x^2-2\cdot4\cdot x+4^2=x^2-8x+16`. The only difference is between the 48 and the 16. To fix this, we can subtract 32 from the 48:

`3(x^2-8x+48-32)`

However, to balance this subtraction out, we'll need to add 96 (which is 32 × 3) on to the end:

`3(x^2-8x+16)+96`

Finally, we can rewrite our function C(x) as

`C(x)=3(x-4)^2+96`

This gives us a vertex/minimum point of (4, 96), which means we need to produce 4 thousand speedboats to minimize its costs.

Answer 2

For minimum cost the number of speedboats produced would be 4,000.

Explanation:

That would be the value of x which minimises the cost C(x).

You can find this by converting the function to vertex form.

C(x) = 3x^2 - 24x + 144

= 3(x^2 - 8x) + 144

= 3[ (x - 4)^2 - 16] + 144

= 3(x - 4)^2 -48 + 144

= 3(x - 4)^2 + 96

For this to be a minimum x must be = 4.

That is 4,000 speedboats.

The actual minimum cost of producing theses is is 96 million dollars.