4 x 2/3 With a number story
2 answers:
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Answer: About 72 feet
The more accurate value is roughly 71.7260397 feet (however, this value isn't fully exact either).
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Explanation:
Check out the diagram below. It's effectively the same diagram your teacher provided, but I've added points M and N. I've left out any unnecessary lines.
- M = center of the circle
- N = location where the cable is anchored to the ground
We're told that "(the length of cable from) E to the ground is 35 feet", so this is the red portion of cable that I've marked this as NE = 35, ie segment NE is 35 feet long. We're also told that NF = 40 feet, which is also shown in red.
The blue portions are chords of the circle. We're given one chord as LE = 80 ft, while the other chord JF is what we want to find out.
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What we need to find is the radius of this circle. Right away, it's not clear what the radius is; however, we can use the chord LE = 80 to help find it.
It turns out that the chord length c is connected to the radius r and central angle theta like so:
c = 2*r*sin(0.5*theta)
where theta must be in radian mode. The central angle subtends the arc that forms the chord in question.
If we started at point L, and counted the number of spaces to get to E while going clockwise, then we can see that there are 5 spaces
Those five spaces could be written as
- The jump from L to A
- The jump from A to B
- The jump from B to C
- The jump from C to D
- The jump from D to E
Or you could mark it as such on the diagram for a visual reference. Overall, the circle has been cut into 12 equal slices. So going from L to E, going clockwise, will have us take up 5/12 of the full circle.
There are 2pi radians in a full circle, meaning the central angle LME is (5/12)*2pi = 5pi/6 radians.
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Plug theta = 5pi/6 and c = 80 into the formula mentioned. Isolate the radius.
c = 2*r*sin(0.5*theta)
80 = 2*r*sin(0.5*5pi/6)
80 = 2*r*sin(5pi/12)
80 = 2*r*0.9659258
80 = 2*0.9659258*r
80 = 1.9318516r
1.9318516r = 80
r = 80/1.931 8516
r = 41.4110483
The radius is approximate. The radius is in feet. Make sure your calculator is in radian mode.
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Now that we know the radius, we can determine how long chord JF is.
The central angle for chord JF is angle JMF
If we start at J and go to F, along the shorter path, then we've gone 4 spaces. This is 4/12 = 1/3 of the full circle. So the radian measure of angle JMF is (1/3)*2pi = 2pi/3 radians.
So,
c = 2*r*sin(0.5*theta)
c = 2*41.4110483*sin(0.5*2pi/3)
c = 71.7260397
Chord JF is roughly 71.7260397 feet long.
When rounding to the nearest foot, that's about 72 feet.
The correct format of the question is
At the end of 2006, the population of Riverside was 400 people. The population for this small town can be modeled by the equation below, where t represents the number of years since the end of 2006 and P represents the number of people. Based on this model, approximately what was the increase in the population of Riverside at the end of 2009 compared to the end of 2006?
(A) 291
(B) 691
(C) 1040
(D) 1440
Answer:
The increase in the population at the end of 2009 is 291 people
Step-by-step explanation:
We are given the equation as
where
P = No of People
t= No of Years
it is given that in the year 2006 the population is 400
this will only happen when we take t= 0
so for
Year value of t
2006- t = 0
2007- t = 1
2008- t = 2
2009 t = 3
No of people in 2009 will be
= 400*1.728
P = 691.2
Since the equation represents no of people so it can't be in decimals, Therefore the population will be 691
Increase = P(2009) - P(2006)
= 691 - 400
= 291
The increase in the population at the end of 2009 is 291 people.