Answer:
Step-by-step explanation:
Distance traveled in hours =
Distance traveled in 1 hour = ÷
÷
distance traveled in 3 1/5 hours =
=
17. Write the standard form of the equation of the circle
1 answer:
Answer:
(x -1)^2 + (y -2)^2 = 13
Step-by-step explanation:
The two given points are the end points of a chord, so the center will be on its perpendicular bisector. That is, the center will be at the point of intersection of the given line and the perpendicular bisector of the given chord.
To find that point, we can write the equation of the perpendicular bisector, then solve the simultaneous equations.
The perpendicular bisector can be written as ...
Δx(x -(3+4)/2) +Δy(y -(5+0)/2) = 0
(3 -4)(x -7/2) + (5 -0)(y -5/2) = 0
-x -9 +5y = 0 . . . . . eliminate parentheses
We can add 3 times this equation to the given equation to find the solution for the circle center.
(3x +2y) + 3(-x -9 +5y) = (7) + 3(0)
17y -27 = 7
y = 34/17 = 2
x = 5y -9 = 1
The circle center is (h, k) = (1, 2).
The square of the radius can be found by substituting one of the given points into the formula for the circle. That formula is ...
(x -h)^2 + (y -k)^2 = r^2
Filling in the values for (h, k) and the first given point, we find ...
(4 -1)^2 + (0 -2)^2 = r^2 = 13
The standard form equation for the circle through the given points with the center on the given line is ...
(x -1)^2 + (y -2)^2 = 13
_____
<em>Comments on equation for a line</em>
There are a lot of ways to write the equation of a line. Often, I like to use standard form: ax +by = c. When given two points, (x1, y1) and (x2, y2), this can take the form ...
Δy(x -x1) -Δx(y -y1) = 0
where (Δx, Δy) = (x2 -x1, y2 -y1).
The perpendicular line through some point (h, k) will be of the form ...
Δx(x -h) +Δy(y -k) = 0
Note the change in sign for the second term and the switching of Δx and Δy. This is what makes the slope be the negative reciprocal of the slope of the above line through the two points.
For the perpendicular bisector, the point (h, k) needs to be the midpoint of the segment between (x1, y1) and (x2, y2). That midpoint is the average of the two segment endpoints:
(h, k) = ((x1+x2)/2, (y1+y2)/2)
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Prove:
____________________________________________
Base Step: For n=1:
and
--------------------------------------------------------------------------
Induction Hypothesis: Assume true for n=k. Meaning:
assumed to be true.
--------------------------------------------------------------------------
Induction Step: For n=k+1:
by our Induction Hypothesis, we can replace every term in this summation (except the last term) with the right hand side of our assumption.
From here, think about what you are trying to end up with.
For n=k+1, we WANT the formula to look like this:
That thing on the right hand side is what we're trying to end up with. So we need to do some clever Algebra.
Combine the (k+1) and 1/2, put the 2 in the bottom,
We want to end up with a 2^k as our final denominator, so our middle term is missing a power of 2. Let's multiply top and bottom by 2,
Distribute the -2 and combine the fractions together,
Combine like-terms,
pull the negative back out,
And ta-da! We've done it!
We can break apart the +3 into +1 and +2,
and the +0 in the bottom can be written as -1 and +1,
____________________________________________
Base Step: For n=1:
and
--------------------------------------------------------------------------
Induction Hypothesis: Assume true for n=k. Meaning:
assumed to be true.
--------------------------------------------------------------------------
Induction Step: For n=k+1:
by our Induction Hypothesis, we can replace every term in this summation (except the last term) with the right hand side of our assumption.
From here, think about what you are trying to end up with.
For n=k+1, we WANT the formula to look like this:
That thing on the right hand side is what we're trying to end up with. So we need to do some clever Algebra.
Combine the (k+1) and 1/2, put the 2 in the bottom,
We want to end up with a 2^k as our final denominator, so our middle term is missing a power of 2. Let's multiply top and bottom by 2,
Distribute the -2 and combine the fractions together,
Combine like-terms,
pull the negative back out,
And ta-da! We've done it!
We can break apart the +3 into +1 and +2,
and the +0 in the bottom can be written as -1 and +1,
a) 20.9
The mean is calculated by adding up the scores and dividing the total by the number of scores, it is also called the average.
First, we can count and see there are 11 players (makes sense for a football team). Adding up the ages gives us 19 + 19 + 20 + 20 + 20 + 21 + 22 + 22 + 22 + 22 + 23 = 230. Divide by 11 and we get a mean of of about 20.9!
b) 30
To figure this out, we will set up an equation that looks like this:
And now we will solve:
[multiply both sides by 12] 230 + x = 264
[subtract 130 from both sides] x = 30
So the age of the new player should be 30!
Have a nice day!
I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)
- Heather