Find an equation of the circle that has center (1, -1) and passes through (5, -6).
1 answer:
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I see you already did the ones where you figured out the formula. There's a fun way that doesn't require figuring out the formula, called a forward difference table.
Let's work out the first one to learn the technique. We write the sequence and under it the forward differences,
2 5 10 17 26
3 5 7 9
2 2 2
We got 3=5-2, 5=10-5, 7=17-10, etc and the same for subsequent lines. Eventually we get to an all constant line and we stop.
To extend the table we just add some constants to the constant line and work our way back up:
2 5 10 17 26 37 50
3 5 7 9 11 13
2 2 2 2 2
So we get 11 as 9+2 and 37 as 26+11 for our answer which matches the answer you already got.
Let's do the ones you haven't.
2 8 18 32 50 72
6 10 14 18 22
4 4 4 4
Answer: 72
3 9 19 33 51 73
6 10 14 18 22
4 4 4 4
Answer: 73
2 11 26 47 74 107
9 15 21 27 33
6 6 6 6
Answer: 107
Three's our limit so I'll leave the last one to you.
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A comment says we need the functions. They're all quadratics. I'll tell you how they want you to do it and how to do it in general.
They just want you to recognize these are all pretty close to just . The one that goes 3 12 27 etc. we recognize as just
. The one that goes 2 8 18 is just
. The one that goes 3 9 19 is one more than that so
. The one that goes 2 11 26... is one less than 3 12 27 so is
. Again I'll leave the last one.
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We can do these quadratics in general. Let's work from our difference table. If we have
f(n) = an^2 + bn + c
and we form the difference table for f(1), f(2), f(3) we get
a+b+c 4a+2b+c 9a+3b+c
3a+b 5a+b
2a
so we can use the first columns of our difference tables to get our polynomial.
2 8 18 32 50 72
6 10 14 18 22
4 4 4 4
Answer: 72
So we have 2a=4, so a=2
3a+b=6, so b=6-3(2)=0
a+b+c=2 so c=2-0-2=0
That gives us the obvious answer
3 9 19 33 51 73
6 10 14 18 22
4 4 4 4
Answer: 73
We have 2a=4 again so a=2, 3a+b=2 so b=0, and a+b+c=3 so c=1 giving
2 11 26 47 74 107
9 15 21 27 33
6 6 6 6
Answer: 107
2a=6 gives a=3
3a+b=9 gives b=0
a+b+c=2 gives c= -1
Answer:
a.
Objective function:
b) See Attached picture
30% in bonds and 70% in stocks
Step-by-step explanation:
a.
In order to solve the first part of the problem we need to take into account that the problem wants us to determine the percentage that should be allocated to each of the possible investment alternatives. In that case, the sum of the percentages must be equal to 1 (which means that we will have 100 of the trust fund)
so that gives us our first restriction for the problem.
B+S=1
Next, the problem tells us that the projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. It also states that he wants to select a mix that will enable him to obtain a total return of at least 7.5%, so we can take this information and get the second restriction from it:
Next, the problem tells us that whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund, so that's where the last restriction comes from:
now, the idea is to optimize the investment, this is get the greatest amount of money out of the trust fund, so the objective function is:
which represents the return on the investment.
b)
For part b we can start by graphing each of the restrictions:
This will be a single line, you can draw the line by setting B=0 first so you get:
0+S=1
S=1
So the first point to plot will be (0,1)
next, we can set s=0 to get:
B+0=1
B=1
so the second point to plot will be (1,0)
so you can plot the two points and connect them with a straight line. This is the green line on the uploaded graph.
Next we can graph the second restriction:
we can use the same procedure we used for the previous graph, in this case the points would be:
(0 , 0.75) and (1.25, 0)
and again connect the two points with a straight line. Next we need to decide which region of the graph to shade for which we can pick two arbitrary points on each side of the line, for example we can pick:
(0,0) and (2,2) and see which one makes the inequality true:
for (0,0) we get:
Which is false, therefore we need to shade the other region of the graph:
for (2,2) we get:
Which is true, so we shade the region of the graph that contains that point. (see red graph)
now we graph the third restriction:
In order to graph this third restriction we just need to draw a vertical line at B=0.3 and shade everything to the right of that line. (Blue graph)
Now, we can analyze the graph, in this case we need to locate the points where the green line crosses the red and the blue line which gives us the following coordinates:
(0.3, 0.7) and (0.625, 0.375)
these two points can be found by setting the first restriction equal to each of the other two restrictions if you are to do it algebraically. If you are using a graphing device, you can directly read them from the graphs.
So once we got those points, we can see which one gives us the greatest percentage of return.
let's test the first point (0.3, 0.7)
R=0.088
so this distribution gives us 8.8% in return, let's test the second point:
(0.625, 0.375)
R=0.075
so this distribution gives us 7.5% in return.
In this case the best distribution for us is 30% in bonds and 70% in the stock fund to get a return of 8.8%
