To All You Mathletes...Part II Quiz

Answer: 1/5

The first guess is a 1/10 probability.

The second guess is conditional on your missing the first guess. Thus it is 1/9*9/10 (the probability of guessing incorrectly the first time). 1/9*9/10 = 1/10.

1/10 + 1/10 = 1/5.

Thanks to Rodney_indy and Wesley Crusher for pointing out the condition on the second guess.

Answer: 24 meters

Your equation is w(w+2)(w-2)=105 which gives w³-4w=105 (where w is the width). Solving for w here gives you w=5. (The other two solutions for w are imaginary, so we can discard them.) Given that the width is 5 meters, the length is 7 meters, and the depth is 3 meters, we know that the perimeter around the top is 2*7+2*5 = 24 meters.

Answer: True

It always works! Try it out with a few different quadrilaterals and you'll see.

The proof is a little hard to explain without diagrams, but it goes generally like this:

Draw the diagonals to the original quadrilateral. By the Midpoint Theory, two sides of the new quadrilateral are parallel to one of the diagonals of the original quadrilateral, and are thus parallel to each other. The same argument applies to the other two sides of the new quadrilateral. Thus the new quadrilateral is a parallelogram.

Answer: y=±abs(x)

One thing to note is that √(x²) is |x|, not just x. A way to see this is to note that no matter what value you plug in for x, x² will always be positive, and the square root of a positive number is positive. The same idea holds true for √(y²). Now the equation looks like this: |y|=|x|. To remove the absolute value brackets from y, we can put a ± sign on the other side of the equation. Doing this yields y=±|x|.

Answer: 6 miles

This is the ambiguous case of the triangle problem. You are given two sides and an angle. The first step to solving these problems is to find the distance to the x-axis from the point 8 miles from the origin at a 30° angle. Call the point where this segment intersects the x-axis C. The x-axis, the 8-mile segment and the distance we are trying to find make a 30-60-90 triangle, so we know that the distance we are trying to find is half of 8, or 4 miles. Now that we know this, we can use the Pythagorean theorem to find the distance from A to C and C to B. Segment AC = CB = √(5²-4²) = 3 miles. Segment AB equals segment AC plus segment CB (because C is between A and B), so segment AB equals 6 miles.

Sorry if that explanation was confusing. It's hard to show without pictures.

Answer: 32π cm³/s

We are looking for dV/dt. We know V(r) and r(t). So we can represent dV/dt as dV/dr * dr/dt. dV/dr is 4πr² and dr/dt is 1/√t. Thus, dV/dt is 4πr²/√t. At t=4, r=4 (from the equation r(t)=2√t). Plugging into dV/dt, we have 4π(4)²/√4 = 32π cm³/s.

Answer: x/ln(x)

x*ln(x) → This is a one-step integration by parts. Integral: (x²/4)(2*ln(x)-1)
ln(x)/x → This is a simple u-substitution. Let u=ln(x) and du=1/x. Answer: ½(ln(x))²
1/(x*ln(x)) → This is a simple u-substitution. Let u=ln(x) and du=1/x. Answer: ln(ln(x))

x/ln(x) → It is impossible to find a formula for this indefinite integral. You can do a numeral approximation if you are given bounds, but without them, this integral is impossible.

Answer: 5 movies, 10 songs

Using Lagrange Multipliers, we can say that the (components of) gradient vectors* of the function and the constraint should be constant multiples of each other. We can set up three equations:

8m^(-0.2)*s^(0.2) = 8λ
2m^(0.8)*s(-0.8) = λ
8m+s = 50

Solving these three equations simultaneously for m and s yields m=5 and s=10.

* The gradient vector is a vector whose components are the partial derivatives of the functions. Ex: Gradient of f(x,y)=xy+y² is [∂f/∂x, ∂f/∂y]=[y,x+2y].

Answer: There is at least one non-zero vector in the kernel of A.

If an nxn Matrix A is invertible, you can say the following:

1. There exists a unique solution x such that Ax=b, where b is a vector.
2. The reduced-row echelon form of A is the identity matrix.
3. The column vectors of A are linearly independent.
4. There are no non-zero vectors in the kernel of A.
5. The image of A spans Rⁿ.
6. The nullity of A is 0.
7. The rank of A is n.
8. The determinant of A does not equal 0.
9. 0 fails to be an eigenvalue for A.

Answer: e^-π/2

If you rewrite i^i as e^(i*ln(i)), you can use the formula for logarithms of imaginary numbers:

Given z = x+iy,
ln(z) = ln(√(x²+y²)) + i*arctan(y/x)

Plugging in from above, ln(i) = ln(√(0+1))+i*arctan(1/0) = 0+iπ/2 = iπ/2.
Now, i*ln(i) = i*iπ/2 = -π/2

Plugging into the original formula, we have i^i = e^(-π/2).

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