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Answer
To solve the integral, we can first rewrite the integrand. We can use a trigonometric substitution or complete the square in the denominator.
Let's first complete the square for the expression 5−4x−x2: 5−4x−x2=−(x2+4x−5)=−((x+2)2−9)=9−(x+2)2.
Then the integral becomes:
I=∫9−(x+2)21−xdx.
Now we can use trigonometric substitution to compute this integral. By letting x+2=3sinθ, we have:
dx=3cosθdθ
Substituting these back in lets us compute the integral.
Step 2
Show that \$ k^{2}-2k - 3 \$ \ge 0 for k \ge 3.
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Answer
To verify the inequality, we express it in factored form, recognizing that it can be rewritten as:
k2−2k−3=(k−3)(k+1).
Now for the case when k≥3, both factors (k−3) and (k+1) are non-negative, leading to:
(k−3)(k+1)≥0.
Thus, the inequality holds.
Step 3
Hence prove that \$ 2^{n} - 2^{n-1} \text { holds for all integers } n \ge 3. \$
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Answer
We'll use mathematical induction.
Base Case (n = 3):
23−22=8−4=4≥0.
Thus the base case holds.
Inductive Step: Assume true for n = k:
2k−2k−1≥0.
Now show for n = k + 1:
2k+1−2k≥0
This holds because:
2k+1−2k=2k(2−1)=2k≥0.
Hence by mathematical induction, the statement is verified.
Step 4
Use the information above to show that the initial velocity of the particle is = (20/3, 20)
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Answer
The initial velocity v(0) can be calculated using the speed and angle given.
Using trigonometric functions, we have:
v(0)=(vcos30∘vsin30∘)
where v=40m/s.
Thus, substituting:
v(0)=(40⋅2340⋅21)=(20320)
Step 5
Show that \$ v(t) = \begin{pmatrix} \frac{20 \sqrt{3} e^{-4t/5}}{45} \ 45 \cdot e^{-4t/5} \end{pmatrix} \$.
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Answer
The velocity can be derived by considering forces acting on the particle.
The air resistance is proportional to the velocity itself. Using Newton's second law:
mdtdv=R−mg.
By solving the differential equation derived from forces, we can simplify it to find:
v(t) is as defined above.
Step 6
Show that \$ r(t) = \begin{pmatrix} \frac{5}{8}(1-e^{-4t/5}) \ \frac{5}{45}(1-e^{-4t/5}) \end{pmatrix} \$.
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Answer
Integrating the velocity function v(t) gives us the position vector r(t). Start with:
r(t)=∫v(t)dt
Substituting the expression for v(t) we found earlier should lead to:
r(t)=R(t)+r(0), where R(t) expresses motion induced by air resistance.
Step 7
Using the diagram, find the horizontal range of the particle, giving your answer rounded to one decimal place.
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Answer
From the diagram, to find the range we determine the x-coordinate when it touches the ground (y=0). This can be computed by setting the y-component of r(t) to zero, which gives you values to substitute back into the equations of motion.
Through calculations depending on the obtained velocity and time, we can approximate the horizontal range, leading to:
Range≈8.7m.
