Question

A QUADRATIC EQUATION QUESTION
The total resources T (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions can be approximated by the equation:

T=-0.26x^2+3.62x+30.18

Where x is the number of years after 2000. Determine when the total resources are at the given level.

1.$30 billion
2.When will the Corporation be out of money (T=0)?

Answer 1

for part 1, solve

\[30 = -0.26x^2+3.62x+30.18 \] for \(x\)

For part 2, solve
\[0 = -0.26x^2+3.62x + 30.18\] for \(x\)

Quadratic formula is probably the way to go:
\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]given \[ax^2+bx+c=0, a\ne 0\]

Answer 2

please can you slove the part 2 for me.

Answer 3

in part 2, we have \(a=-0.26, b = 3.62, c = 30.18\) so our solutions are
\[x=\frac{ -3.62\pm\sqrt{3.62^2-4(-0.26)(30.18)}}{2*(-0.26)} =\]

Answer 4

oh ok.got you now you so good.can i ask you for another help with another question?

Answer 5

what's the question?

Answer 6

A well-known golfer sells his book Help of Hackers for $13.25 per copy. His fixed costs are $125,000 and he estimates that the variable cost of printing, binding, and distribution are given by

-4x^2+2880x-295.

Find expressions for the revenue, cost, and profit from selling x thousand copies of the book.

Answer 7

revenue is sales, so revenue = price per copy * x * 1000 (because x is in thousands of copies)

cost is fixed cost + variable cost, and those are both given to you

profit is revenue - costs, so just combine the two previous equations accordingly

Answer 8

so u mean the revenue = $13.25x(thousand)

Answer 9

The price per book is $13.25, but x=1 is 1000 books, so the revenue from x=1 would be $13250.

Answer 10

so revenue = 13250x

Answer 11

so can u please write out the cost for me.?want to see if im right

Answer 12

you write out what you think it is, and I'll tell you if you have it right.

Answer 13

$125000-4x^2+2880x-295

Answer 14

Yes.

Answer 15

so profit =13250x-125000-4x^2+2880x-295 or ?? i guess im wrong

Answer 16

well, regardless of any question of whether you've added/subtracted the right pieces, you did the algebra wrong...

\[revenue = 13250x\]
\[cost = 125000-4x^2+2880x-295\]
\[profit = revenue - cost = 13250x - (125000-4x^2+2880x-295)\]\[=13250x-125000-(-4x^2)-2880x-(-295)\]

Answer 17

when you subtract a polynomial, you have to put parentheses around the whole thing and distribute that -1 factor...

Answer 18

\[x+a - x -b = a-b\]\[x+a - (x-b) = a+b\]

Answer 19

so after this,need i group like terms and subtract or add them accordingly or i should just leave the answer like that?

Answer 20

oh, I would collect like terms and simplify, definitely!

Answer 21

just make sure you don't make any mistakes in the process :-)

Answer 22

Here's a graph of cost, revenue and profit. Can you figure out which line is which? How many thousand books need to be sold (x=) before there is a profit?

Answer 23

wow you are soo helpful.

Answer 24

Note the fact that the cost equation is a parabola makes pretty much no difference at all, because the constant factors are >> the value of x^2 for the range of interest.

Answer 25

it's only when sales get out into the hundreds of thousands of copies that you can see it on the graph:

Answer 26

were you able to figure out which line is which, and estimate the break-even point for selling enough copies to start making a profit?

Answer 27

so after simplifying i need to make sure my x^2 comes first followed by my x number and then the number without any variable.just like how it appears in the variable cost?

Answer 28

well, I don't think you have to write it in that order, but usually people do (or in the reverse order).

Answer 29

oh ok.that would therefore be my final answer right after i have simplified right?

Answer 30

Yes. Though to be honest, even without simplification or rearrangement you've satisfied the request to "Find expressions for the revenue, cost, and profit from selling x thousand copies of the book."

Answer 31

oh kk.that would be all then .thank you very much you far too kind.im very grateful :-)

Answer 32

If they want fully simplified expressions written in descending order (or the problem is marked incorrect), the problem statement should say so :-)

Answer 33

You're welcome. I like to show people that there are actually interesting things going on in these problems besides just finding the solution :-)

Answer 34

:-)

Answer 35

For example, you can see in that cost graph that it costs a bit less to make each additional book once you're making a sufficient volume of them. volume discounts, baby!

Answer 36

hahhahahahaha.i gotcha