If the circumference of a circle is 36, what is the length of an arc of the circle intercepted by a central angle of 30 degrees?
2 answers:
Answer:
Step-by-step explanation:
The formula for calculate the arc lenght is:
Where "r" is the radius and "" is the central angle of the arc in degrees.
The formula used to find the circumference of a circle is:
Where "r" is the radius.
Then, we can observe that the formula for calculate the arc lenght can be rewritten in this form:
Where "C" is the circumference of the circle.
Finally we need to substitute the central angle and the circumference into . Then the result is this:
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and $\theta$ lies in $2^{\text{th}}$ quadrant.
where, $x-$ coordinate is negative, and $y-$ coordinate is positive
so it can't a.
now, cosine means, side adjacent over the hypotenuse, in Cartesian plane, that will be $x-$ coordinate over the distance from origin.
Assume the triangle , with base $2$ units and hypotenuse $\sqrt{29}$ and it's in second quadrant. (so )
now, the leftmost point on $x-$ axis is , obviously $(-2,0)$
and by Pythagoras theorem, we can find the perpendicular side, that will be $y^2=(\sqrt{29})^2-(2)^2\implies y=5$
so the coordinates of the upper vertex is $(-2,5)$, each point lying on this "ray" should have equal ratio of respective coordinates. i.e. $\frac25=\left|\frac xy\right| $
and it should lie on second quadrant, so $x<0 \, y>0$
Option d satisfies this.
See the attached image to see how all of the answers fill out.
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Problem 6a
The slope of line A is -1/1 or just -1
The reason why is because we start at (0,3) and move down 1 and over to the right 1 to get to the next point (1,2). Alternatively you can use the slope formula m = (y2-y1)/(x2-x1) to get the same result.
Since the slope of line A is -1, this means that we'll write "-1" in the first row of the "constant rate of change" column.
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The two points (0,2) and (1,3) are on the line B
Use the slope formula to get
m = (y2-y1)/(x2-x1)
m = (3-2)/(1-0)
m = 1/1
m = 1
The slope of line B is 1. Write "1" in the second row of the "constant rate of change" column.
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The points (0,-4) and (1,-2) are on line C
Slope:
m = (y2-y1)/(x2-x1)
m = (-2-(-4))/(1-0)
m = (-2+4)/(1-0)
m = 2/1
m = 2
The slope of line C is 2, which is written in the last row of the "constant rate of change" column.
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Now onto the y intercept column.
Line A crosses the y intercept at the point (0,3) or simply 3 on the y axis. So we'll have 3 in the first row.
Write 2 in the next row under the "3" to indicate that the y intercept of line B is 2. The black line B crosses the y axis at 2 on the vertical number line.
Similarly, the y intercept for line C is -4. This red line crosses the vertical y axis at -4
-----------------------------------------
Now onto the x intercept
The x intercept of line A is 3. This is where the blue line crosses the horizontal x axis.
The x intercept of line B is -2. This is where the black line crosses the horizontal x axis.
The x intercept of line C is 2. This is where the red line crosses the horizontal x axis.
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Problem 6b
Equation D: y = 2+x
is the same as
y = x+2
The slope here is 1 (since y = x+2 is the same as y = 1x+2) and the y intercept is 2.
The fact that
Slope = 1
and
y intercept = 2
means we have a match with line B
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The equation y = -4+2x is the same as y = 2x-4
Slope = 2
y intercept = -4
this matches with line C
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y = 3-x is the same as y = -x+3
which can be written as y = -1x+3
Slope = -1
y intercept = 3
This matches with line A
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See the attached image to see how all of the answers fill out.
