9514 1404 393
Answer:
t ∈ {19, 85, 109, 175, 199}
or
t ∈ {40, 64, 130, 154, 220}
Step-by-step explanation:
A graphing calculator relieves the tedium of solving for t. The attachments show the times the satellite is 4000 miles from the equator. Since y may be either north or south, the satellite may be 4000 north of the equator for y = 4000 or for y = -4000.
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If y is miles <em>north</em> of the equator, then ...
4000 = 6000cos(z) . . . . . where z=(π/45)(t-7)
z = arccos(4000/6000) ≈ 0.84107 radians
There is another solution at 2π -0.84107 radians, about 5.44212 radians
The corresponding time values are ...
t = 7 +(45/π)(z) ≈ 19.047 and 84.953 . . . . minutes
The period of the function is 90 minutes, the times will be the above times and at 90-minute intervals after each.
The satellite is 4000 miles north of the equator at ...
t ≈ 19, 85, 109, 175, 199 . . . minutes after launch
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The solution is similar if y represents miles <em>south</em> of the equator. Then we have ...
-4000 = 6000cos(z)
z = arccos(-4000/6000) ≈ 2.30052 radians (and 3.98266 radians)
The corresponding time values are ...
t = 7 +(45/π)(z) ≈ 39.952 and 64.047 . . . . minutes
Again, considering the period, the satellite is 4000 miles north of the equator at ...
t ≈ 40, 64, 130, 154, 220 . . . minutes after launch
_____
<em>Additional comment</em>
We might usually think of y as being positive in the northerly direction. However, in this age of political correctness and bias sensitivity, we have to consider that "y is north or south" means exactly that. So, we have also shown the solutions when y is positive south of the equator.
Usually, we prefer that the variables are defined unambiguously.
Answer:
Step-by-step explanation:
The discriminant is found in
which is part of the quadratic formula. In
, a = 4, b = 10, c = -16. Filling in for the discriminant:
It's positive so we know we have real roots; it's not a perfect square, so our solutions are 2 complex rational.
In , a = -16, b = -7, c = 3. Filling in for the discriminant:
It's positive and not a perfect square so the 2 solutions are complex rational
Answer:
The sum of the first 20 terms is -1440.
Step-by-step explanation:
We want to find the sum of the first 20 terms of the arithmetic sequence:
4, -4, -12, -20...
The sum of an arithmetic sequence is given by:
Where <em>k</em> is the number of terms, <em>a</em> is the initial term, and <em>x</em>_<em>k</em> is the last term.
Since we want to find the sum of the first 20 terms, <em>k</em> = 20.
Our initial term <em>a</em> is 4.
Our last term is also the 20th term as we want the sum of the first 20 terms.
To find the 20th term, we can write an explicit formula for our sequence. The explicit formula for an arithmetic sequence is given by:
Where <em>a</em> is the initial term, <em>d</em> is the common difference, and <em>n</em> is the <em>n</em>th term.
Our initial term is 4. From the sequence, we can see that our common difference is -8 since each subsequent term is eight less than the previous term. Therefore:
Then the last or 20th term is:
Therefore, the sum of the first 20 terms are:
According to the secant-tangent theorem, we have the following expression:
Now, we solve for <em>x</em>.
Then, we use the quadratic formula:
Where a = 1, b = 6, and c = -315.