00:01
Carol has 2 ,400 feet of fencing to fencing a rectangular horse coral.
00:08
In part a, we will find a function that models the area of the coral in terms of the width x of the coral.
00:17
In part b, we find the dimensions of the rectangle that maximize the area of the coral.
00:24
So we have a rectangle.
00:30
The coral is a rectangle.
00:34
Let's say this.
00:34
That has, let's say long or length y and width, and width x like this.
00:49
And we have 90 degrees in each corner.
00:58
Okay, that's our coral and mathematically or geometrically is a rectangle with high, with x and length y.
01:12
So we know that the area of the rectangle is given by x times y.
01:26
And we want to write the area of the coral, that is the area of the rectangle, in terms of the width of the coral, that is in terms of this variable here.
01:42
And we have to write a relationship between the two variables in order to find y in terms of x, for example, and put it in this equation.
01:53
And that can be done through the fact that we have a number of feet of fence, of fencing to fencing the rectangular horse corals.
02:08
So we have this quantity of fencing.
02:13
And we know that the fencing, the quantity of fencing corresponds to the perimeter of this rectangle.
02:22
So the feet of fencing is equal to the perimeter.
02:39
Of the rectangle.
02:45
It means that the perimeter is 2x plus 2y because we get to sum up all the sides of the length of the sides of the rectangle and we get 2x and 2y.
03:04
And that must be equal to the number of feet of fencing we have.
03:10
That is 2400.
03:15
So this equation means there is a relationship between x and y.
03:21
Remember x and y get to say here must be given in fit in order for all units to be coherent.
03:31
So we get this equation and from this we get x plus y divided by two both sides is 1200 and now we can solve for y for example why is 1200 minus x and this equation, we are going to use to plug in into the area equation.
03:57
When we do that, we get that a is equal to x times y, which is replaced by 1 ,200 minus x.
04:15
And this is the area which is 1 ,200x minus x square.
04:21
So a of x, formally speaking, there is the area in terms of the width, of the rectangular coral is 1200x minus x square and this is a parabola as you can see.
04:43
It's a parabola that opens downwards because the coefficient of x squared is negative, it opens downwards and with roots we already know them.
05:07
Zeros, maybe it's better.
05:13
It's zeros.
05:16
X equals zero because we have this factorization already of the function, that is x times this other factor...