Problem of the Week Day 2: Week of 6/17/13 - 6/21/13

Today is day 2 of the problem of the week! Remember to substitute in all of yesterday's answers into today's problems.

Tuesday is the day that I set aside for Algebra. Since the easy level does not use actual material from an Algebra course, I have included things that students need to know for an algebra course (fractions and decimals, order of operations, etc.). I have also tried to make the arithmetic challenging in that section to make the actual task a little harder than normal.

Easy:
First, calculate the value of m and n using the values of a and p. Reduce the fractions once you have found them.

        2617736

m = –––––––

         20775a

      5516
n = ––––
      277p


Now, solve for y and z using the rules for subtracting and dividing fractions.

y = m - n
z = m ÷ n

m = _____
n = _____
y = _____
z = _____

Medium:
Substitute the values of p and h into the following equation, and then solve for the value of t.

p(ht + 705) = (3t - 56)(6h - 2p)³

t = _____

Hard:
First, find the value of q₀ using the value of p from yesterday.

       9p + 47

q₀ = –––––––

            2

Now, q₀ is the zeroth term in a geometric sequence. The recursive formula for this sequence is the following:

qn = (½)qn-1

You can find values of this sequence now or at a different time, but you may need up to the 15th number in this sequence. Be prepared to be referring back to this many times throughout the week.

Knowing the values of the geometric sequence, simplify the following rational function (you can leave it factored, but make sure x is the only variable):


                   (x - q₁₁)(x - q₉)
f(x) = ––––––––––––––––––––––
          (q₁₁x + p)(x - q₁₀)(x + q₁₁)

Now, determine the vertical and horizontal asymptotes of this function. Use x₁, x₂, x₃ and so on for the equations for the vertical asymptotes (if the asymptotes were 2 and 5, but x₁= 2 and x₂= 5), and y₁, y₂, y₃ and so on for the equations for the horizontal asymptotes. Make sure that the asymptotes are arranged from least to greatest in relation to the variable subscripts.

I am not going to put down the variables to find with the blank. Just find as many values of q, x, and y that exist or are necessary for the problem.