00:01
Okay, here we have the coral which is in the shape of a rectangle with the semicircle and the diameter of the semicircle is the side of the rectangle as shown in this diagram.
00:14
So this x represents the diameter of the circle as well as the side of the rectangle.
00:20
And we are told that to enclose this coral, 201 feet of fencing is required.
00:26
And in this case, we have to determine the maximum area and we have to determine the corresponding dimensions which queues the maximum area.
00:36
That is, we have to determine the side length x as plus y, which gives the maximum area of this coral.
00:44
So let's see how to determine this.
00:47
So it basically involves about three steps.
00:51
The first step is to write the fencing equation.
00:54
We will write the fencing equation and then in the second step we have to write the area equation area equation for the coral and in the third step we'll reduce calculus to maximize the area that is at what point of x or y the area is maximum and using that we can determine the other side line so first we'll do the first step that is we have to write the fencing equation.
01:27
Since we are given that 201 feet is the maximum fencing, let's set up the fencing equation now.
01:35
So as you can see from this diagram, this side length is x and this side length is y.
01:41
So the fencing involves we have to consterned this x and then plus y and so basically we have 2y.
01:52
So we can put this as 2y.
01:54
We have 2y.
01:55
We have 2.
01:55
We have are two side lengths whose dimension is y.
01:58
So i'm going to put x plus to y.
02:01
So this is the fencing which is required to fence along the rectangle sides of the rectangle and now we have to fence along the semicircle.
02:13
We know that if we consider a circle, the perimeter or the circumference of the circle is 2 pi times radius.
02:23
However, if it is for a semi -circle, circle, we have to divide this by 2, so which gives pi r.
02:29
So the pincing, the fencing, which is required for a semicircle, if we do not consider this, as far as this coral is considered, we are not considering this side length when we consider the semicircle.
02:43
So it's basically only pi times and the radius.
02:47
And since the side length or the diameter of the semicircle is x, we say that the radius of the semicircle is x by 2 and we will utilize this this is the fencing which is required for the semicircle so therefore the fencing of the semicircle is pi times of x by 2 and this gives pi x by 2 we will add this with our fencing equation that is pi x by 2 and this equals the total fencing which is required to enclose the score which is 201 so we have set up this equation that is x plus 2y plus pi x by 2 this equals 201 from this i'm going to solve for y so that i can substitute when i find the area equation so first let me solve for 2y and this equals 201 minus x minus pi x by 2 now i have to divide both sides by 2 so that i solve for y so therefore y equals 201 divided by 2 minus of x by 2 minus pi x divided by 2 minus pi x divided by 2 and then multiplied by 1 by 2.
03:59
So this equals pi by 4.
04:01
I can multiply this 2 in the denominator and i can simply write this as pi x by 4.
04:07
We can write this in simplified form.
04:11
So let's do that.
04:12
So we get to y equals 201 divided by 2 is 100 .5.
04:18
And we can factor x by 2 from these two terms so i'm going to factor minus x by 2 and when i do that i get 1 plus pi by 2 we can double check this so when we multiply this minus x by 2 times 1 we get minus x by 2 and minus x by 2 times pi by 2 is minus pi x by 4 which we have it here so this factorization is correct so this is the form that we will utilize for y.
04:52
We are going to utilize this form of y.
04:55
That is 100 .5 minus x by 2 times 1 plus pi by 2.
05:02
Okay, so now we will do the second step.
05:05
That is we are going to write the area equation.
05:08
So we have to find the area of this coral.
05:11
This area of this coral and this involves area of the rectangle plus area of the semicircle.
05:31
We know that the area of the rectangle is length times width.
05:35
So here we can consider this x as length or y as width.
05:39
So therefore it is xy plus area of the semicircle.
05:44
We know that area of a circle is pi times radius square.
05:49
And area of semicircle we have to divide this by 2.
05:52
So it's basically pi times radius square by 2.
05:55
And we already know that radius of the semicircle is x by 2, which we will substitute here and replace the radius.
06:05
So therefore this becomes xy plus we can write down this as pi by 2.
06:10
R is x by 2.
06:11
I'm replacing r as x by 2 quantity squared.
06:16
So this gives x times y plus pi by 2 times x by 2 quantity square is x squared by 4.
06:25
Fact we can simplify this so this becomes pi x squared by 8 so i can just replace this as pi x squared by 8 we will also replace y now since we have to write down this area in the one variable we cannot work with the two variables so we have to replace y in terms of the expression for y as we have determined after step one so let's do that and when we do that we can consider this as a function of x so i'm replacing a by a of x so this becomes x times of let me replace y as we have found in the first step that is 100 .5 minus x by 2 times 1 plus pi by 2 and then we have this pi x squared by 8 so i'm writing this pi x squared by 8.
07:30
Now let's get this simplified.
07:33
So we have a of x.
07:35
I'm going to distribute this x into the terms inside the brackets.
07:39
So x times of 100 .5 is 100 .5x minus x times of x by 2 is x squared by 2 times we have this factor 1 plus 5 by 2.
07:52
We also have the last term that is 5x squared by 8.
07:57
We can also simplify this further by factoring out x squared from these two terms.
08:05
So let me do that.
08:06
So this becomes 100 .5x plus i'm factoring the x squared.
08:15
So when i factor the x squared first let me consider this term.
08:20
I have pi by 8.
08:22
So i put pi by 8.
08:24
Now i'm factoring only x squared from this term.
08:27
That is the first term will be minus 1 by 2 times 1 if you do the distribution.
08:35
So i have to put this minus 1 by 2 and the next term will be minus 1 by 2 times pi by 2 which is minus pi by 4.
08:45
So these are the factors that we get after we factor the x squared.
08:50
Now we will simplify this.
08:52
So a of x and this equals 100 .5.
08:57
5x plus x squared pi 2 we can combine these two terms and simplify this so let me do it here this is pi by 8 minus pi by 4 to make common denominator i multiply the second term by 2 and divide by 2 so therefore this becomes pi by 8 minus 2 pi by 8 so therefore we can now add the numerators pi minus 2 pi is negative pi by 8.
09:31
So when we combine these two terms, when you simplify this as a single fraction, we get negative pi by 8.
09:38
And we already have negative 1 by 2.
09:41
So once again, we can also simplify this by factoring out the negative.
09:45
So therefore, i can write down this area of function.
09:49
And this equals 100 .5x minus i'm just factoring on only the negative.
09:56
So this will become negative x squared by 2.
10:00
Times of 1 by 2 plus pi by 8...