Complex numbers. Solve the following, writing the final answer in a+bi form.

| August 30, 2017

Question
(Section 3.1) Complex numbers. Solve the following, writing the final answer in a+bi form.
For part e), there should be no i -term in the denominator, as in, 3/(8-97i) is wrong – note the red i. An i-term in the numerator, like (14-12i)/17 is good.

√(-98)
√(-3)∙√(-18)
i(-3+2i)
(-1-7i)∙(3+9i)
2i/(2-3i)

(Section 3.2) The following is an equation “reducible to a quadratic,” and can be factored into a product of four linear factors like 6x-9:

x^4-5x^2+4

Factor the above polynomial into four linear factors.
What are the roots, as in, which numbers solve x^4-5x^2+4=0?

3) (Section 3.3) Given 72 feet of fencing a garden is to be enclosed in one of three ways:

It can be enclosed as a square.
Or, as a circle, see left for important formulas,
Or, as a rectangle, with a 15-foot shed as one of its sides.

Here is the real problem: You have 72 feet of fence for a garden. You wish to enclose the max area possible, fenced to keep deer out.

As to using the shed as a side, of course, it isn’t that simple. The shed’s wall is 15 feet long. So, what is the garden of maximum area that can be enclosed of:

A rectangle, with the 15-foot shed one side,
A square
A circle, (for full credit, round to the nearest tenth of a square foot)

with the following constraints?

There is 72 feet of fencing, all of which is to be used and
Of a rectangular garden, the shed wall may be one side. It is 15 feet long.
As much room as is needed is available, should you wish to enclose a square or a circle.

Learn anything?

4) (Section 3.4) Solve, or state that no solution is possible:

∛(2x-1)=|-2|

5) (Section 4.1) The book tells you how to find the roots (the zeros) of a quadratic polynomial. They don’t tell you, but it’s true, a polynomial’s maximum (or minimum) value is occurs at the x-value exactly halfway between the two roots – their midpoint.

This trick does not work for cubic polynomials, quartic polynomials, in fact, for any polynomial of order higher than two.

You are playing soccer at an indoor soccer complex. The roof’s height at the facility is 75 feet.

During a game, if a kicked soccer ball touches the ceiling, the player that kicked the ball must sit out for two minutes. From the dirt, you kick a ball that goes straight up with an initial velocity of 70 feet per second.

This means the ball’s height is H is given by the equation H(t)= -16t^2+70t, where t is time.

Does the ball hit the ceiling?

6) (Section 4.5) Don’t waste time graphing the function, which will get zero credit. Use algebra to determine the vertical asymptotes of

(x^2+1)/(2x^2+x-1)

7) Section 5.2) An isotope (if you don’t know the meaning of the chemistry-physics word “isotope,” don’t worry about it) of carbon is carbon-14. Carbon 14 is radioactive, decays with a half-life of 5720 years, as in: a gram of carbon 14 today decays, and in 5720 years, only a half-gram survives.

Given an initial value of 20 grams of carbon 14, will hand you the equation of how many grams C(t) of carbon-14 survive at year t:

C(t)= 20e^(-0.00011211t)

You have a sample of 20 grams of carbon-14. How many years will it take to decay so that only 5 grams is left?

This problem is as non-linear as the wind in a storm. Drop all linear methods that worked on other problems. The only sensible way is through the use of natural logarithms.

Round to the nearest year. Don’t worry about months or days.

A tip for such problems: Carry all digits until the bitter end. Round the final answer, but nothing before, as “Small errors can really add up.”

Now see why carbon-14 is used to assigned dates to really old things? Your body, my body – we are have trace quantities of carbon-14, consequently are radioactive. The potassium-40 in our bodies is also radioactive, but like carbon-14, it is a mild, mild radioactivity, and potassium-40 occurs in only trace quantities. Don’t let the fact that you are radioactive keep you awake tonight.

8) (Section 5.2) You are practicing jumping rope. Each day, you do so many successful jumps before catching on the rope. On a plus note, each day you double the number of successful jumps you did the day before.

The function, J(d), for jumps per day, is J(d)= 2∙2^d.

Problem:

How many successful jumps will you be able to do, on day 12?
Do exponential functions grow, really, really fast?

9) (Section 5.3) A service-member handed you a logarithm, then passed out, cold. In base-ten, the value is

log⁡(M)= 0.2561

You need two things, and fast, to stop an enemy missile from plowing into your base:

That logarithm, but as a natural logarithm, that is, in base e.
The number M.

Your commanding officer has told you, “I need these numbers, to the nearest thousandth, in two minutes. That’s all we have.” Can you do it?

As usual, do no rounding until the final step. Errors can really add up.

10) (Section 5.3) Earthquake intensity is measured by the Richter scale. The formula for the Richter rating R of a given quake is given by R = log (I/I_0 ) where I_0 is the “threshold quake”, or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.

How many times〖 I〗_0 is a 9.0 earthquake? Let the intensity of a 9.0 earthquake be the variable I_9.
How many times〖 I〗_0 is a 6.0 earthquake? Let the intensity of a 9.0 earthquake be the variable I_6.
How many times more powerful is a 9.0 earthquake than a 6.0 earthquake?

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