Answer:
4 thousand speedboats
Step-by-step explanation:
<h3>Vertex Form</h3>
The minimum/maximum point on a parabola is just another name for that parabola's <em>vertex</em>. A <em>parabola </em>can be defined in a few different ways, but one is as the curve described by a <em>quadratic function</em>, a function of the form
where a, b, and c ≠ 0. To see how we can get a vertex out of this, we can start with the simpler function
. 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
and the vertex (h, 0), and if we shift it vertically by k units, we get the equation
and the vertex (h, k). We can, of course, add k to either side to obtain the function
, also known as the general <em>vertex form </em>of a quadratic function.
<h3>The Problem: Completing the Square</h3>
This problem asks us to find a value for x which would <em>minimize </em>the C(x) in the function
. 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 <em>completing the square </em>to transform the expression on the right side into the form we want. Our task then is to somehow manipulate
so that it resembles the form
, 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)](https://tex.z-dn.net/?f=3x%5E2-24x%2B144%5Crightarrow3%28x%5E2-8x%2B48%29)
What we'd like now is to somehow turn that expression in the parentheses into something resembling
. To do this, we can recall that
.
If we rewrite
as
, we can see that this <em>almost </em>resembles
. 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)](https://tex.z-dn.net/?f=3%28x%5E2-8x%2B48-32%29)
<em>However</em>, to balance this subtraction out, we'll need to <em>add </em>96 (which is 32 × 3) on to the end:
![3(x^2-8x+16)+96](https://tex.z-dn.net/?f=3%28x%5E2-8x%2B16%29%2B96)
Finally, we can rewrite our function C(x) as
![C(x)=3(x-4)^2+96](https://tex.z-dn.net/?f=C%28x%29%3D3%28x-4%29%5E2%2B96)
This gives us a vertex/minimum point of (4, 96), which means we need to produce 4 thousand speedboats to minimize its costs.