Let
x-------> the time
y-------> the altitude of a helicopter
Step 1
<u>Find the slope of the line</u>
we know that
the slope m of the line between two points is equal to
we have
substitute in the formula
therefore
<u>the answer is the option </u>
The slope is . This means that the helicopter descends ft each minute.
Notice that
So as
you have
. Clearly
must converge.
The second sequence requires a bit more work.
The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then
will converge.
Monotonicity is often easier to establish IMO. You can do so by induction. When
, you have
Assume
, i.e. that
. Then for
, you have
which suggests that for all
, you have
, so the sequence is increasing monotonically.
Next, based on the fact that both
and
, a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.
We have
and so on. We're getting an inkling that the explicit closed form for the sequence may be
, but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.
Clearly,
. Let's assume this is the case for
, i.e. that
. Now for
, we have
and so by induction, it follows that
for all
.
Therefore the second sequence must also converge (to 2).
Once again, 1200 would be the correct anwser
+
= 1
This is because the standard form of an ellipse is
+
= 1
where a is the vertex and b is the co-vertex. So when we stick their respective x and y values in and then square them, you're left with the answer above.
1,325-35=1,290
50x25=1,250 50x26=1,300
25 months