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Math 3
Math 3  
Hello again. Just back from a fortnight in Florida and gradually picking up where I left off. As promised, here's the first of another pair of sets of geocaching related mathematical questions. (The second set are a lot harder!) I've taken a screen-shot and attach it as an image to ensure you see what I'm seeing.

I have managed some of them for myself:-
1. x y = 3   x + y = 5  x = 4 y = 1  h=1
2. 2x^3 + 3x^2 - 8x + 3     c=3
3. k=5  (I stuck the equation into Wolfram and estimated from the diagram it drew!)
4.
5.
6. 7^x = 108.81  x = 2.40998 = 2.41  f=4
7.
8.
9. 0.6 + (0.4 * 0.8) = 0.92  b=9
10. e=1

Answer
1. Equating exponents

You are exactly right in setting up and solving this system.

The only issue is that you have the wrong one. I think you want h=x=4, not h=y=1.

2. Taking derivatives

There are a number of ways to do this, but you are missing a key step which is to take the derivative (gradient). The equation you have doesn't match the form mx^2 + nx + c.

If you take the derivative you get 6x^2 + 6x - 8, so c = -8.

3. Completing the square

This expression can be rewritten as:

(x-3)^2 - 9 + (y-5)^2 - 25 = 0

or simply:

(x-3)^2 + (y-5)^2 = 34

To the nearest integer, the radius is √(34) = 5.83095... ≈ 6

4. Integration

This is just a reverse of problem 2. You have the derivative dy/dx = 8x-5.

This means y = 4x^2 - 5x + [C], where [C] is a constant of integration.

Based on the point (3,4) being a point on the curve, you get:

y = 4x^2 - 5x - 17

5. Trapezoid Rule

This is really just a formula:

f(x) = log[10](3+x)

I ≈ (1/2) [ f(4) + 2 f(4.5) + 2 f(5) + 2 f(5.5) + f(6) ] / 2

This gives you 1.80362... (which is accurate to four decimal places).

6. Logarithms

This is really just the literal definition of the logarithm:

x = log[7](108.81) &asypm; 2.40998

7. Roots of polynomials

Complex roots of polynomials always come in conjugate pairs. The other root is 9+5i, so I guess they think j=9.

8. Acceleration

Frankly, there are only two conditions on this motion:

The final velocity is zero.

The final displacement is 6.

There are three unknowns:

Her acceleration.

Her deceleration.

When she stopped accelerating and started decelerating.

This problem cannot be solved with more information. If you assume it's halfway, the answer is 12, but I'm not a huge fan of that idea.

The other possibility is that there is a typo and the question really means to say that the "timer" of 3 seconds starts not from her resting position but from when she starts breaking. In that case, her deceleration is known to be enough to slow herself down from the initial velocity in 3 seconds, while traveling 6 meters. That would give deceleration as 4/3 and velocity as 4.

So try 4 and 12 and see if that works. Otherwise this question is just bad.


9. Probability

Well, first is the 0.6 chance that he hits it.

If he misses (0.4 chance of miss), that gives an overall chance of a second hit of 0.40.8 = 0.32.

His total chance is 0.6 + 0.32 = 0.92.

10. Logarithms

Despite the horrible "abuse of notation," the question wants the natural logarithm of the natural base e, which is 1, just like the base-10 log of 10 is 1. You seem to know this.

However, "e" in the question also means the intended result, so e=1... confusing.

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Clyde Oliver

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I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks.

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