Miguelle is deciding which type of heating system to put in his house. An oil system costs $2700 to install and
1 answer:
Answer: (9, 10800)
Step-by-step explanation:
We will first let x be the number of years and y be the total cost.
In that case, let's plug in the values for each into the equation <em>y=mx+b, </em>where m is the slope and b is the y-intercept.
The y-intercept will be the <u>value we start with before any years pass</u>, so it will be the <u>installation cost</u>. The slope is <u>how fast the total cost will increase</u>. Since the <u>operation costs</u> have to be added on every year, it will be our slope.
With that in mind, let's create both of our equations:
- y = 900x + 2700 (Oil system)
- y = 200x + 9000 (Solar system)
Since y is solved for in the first equation, we can substitute it into the second equation:
<em>[Subtracting 200x from both sides]</em>
<em /> <em>[Subtracting 2700 from both sides]</em>
<em /> <em>[Dividing both sides by 700]</em>
<em />
We can now substitute 9 for x in any equation and solve for y. Here I substituted it into the first one:
<em>[Multiplying]</em>
<em /> <em>[Adding]</em>
Hence, the solution to this linear system is (9, 10800).
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Answer:
Summation notation:
or after using your function part:
After evaluating you get 11.125 square units.
Step-by-step explanation:
The width of each rectangle is the same so we want to take the distance from x=0 to x=3 and divide by 6 since we want 6 equal base lengths for our rectangles.
The distance between x=0 and x=3 is (3-0)=3.
We want to divide that length of 3 units by 6 which gives a length of a half per each base length.
We are doing right endpoint value so I'm going to stat at x=3. The first rectangle will be drawn to the height of f(3).
The next right endpoint is x=3-1/2=5/2=2.5, and the second rectangle will have a height of f(2.5).
The next will be at x=2.5-.5=2, and the third rectangle will have a height of f(2).
The fourth rectangle will have a height of f(2-.5)=f(1.5).
The fifth one will have a height of f(1.5-.5)=f(1).
The last one because it is the sixth one will have a height of f(1-.5)=f(.5).
So to find the area of a rectangle you do base*time.
So we just need to evaluate:
or by factoring out the 1/2 part:
To find f(3) replace x in -x^2+2x+4 with 3:
-3^2+2(3)+4
-9+6+4
1
To find f(2.5) replace x in -x^2+2x+4 with 2.5:
-2.5^2+2(2.5)+4
-6.25+5+4
2.75
To find f(2) replace x in -x^2+2x+4 with 2:
-2^2+2(2)+4
-4+4+4
4
To find (1.5) replace x in -x^2+2x+4 with 1.5:
-1.5^2+2(1.5)+4
-2.25+3+4
4.75
To find f(1) replace x in -x^2+2x+4 with 1:
-1^2+2(1)+4
-1+2+4
5
To find f(.5) replace x in -x^2+2x+4 with .5:
-.5^2+2(.5)+4
-.25+1+4
4.75
Now let's add those heights. After we obtain this sum we multiply by 1/2 and we have our approximate area:
Okay now if you wanted the summation notation for:
is it
or after simplifying a bit:
If you are wondering how I obtain the .5+.5(k-1):
I realize that 3,2.5,2,1.5,1,.5 is an arithmetic sequence with first term .5 if you the sequence from right to left (instead of left to right) and it is going up by .5 (reading from right to left.)