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#### needalgebra

##### Member
This is what i got...

For option 1 I think it's y=( .5(x-20))^2+20 for x>20

For option 2 I think it's y=30+.5*ln|x-25| for x>25

Question:

develop equations that model the two replacement pack options

(option 1) quedratic model, monthly access fee \$20, included gigabytes 20, cost per additional gigabyte *see below (*) (option 2) logarithmic model, monthly access fee \$30, included gigabytes 25, cost per additional gigabyte *see below (**)

(*) The charge option 1 will increase quadratically after 20 gigabytes with customers paying \$0.50. (**) The charge option 2 will increase logarithmically in the form ƒ(x) = a + b 1n|x| after 25 gigabytes with customers paying \$0.50.

Last edited:

#### needalgebra

##### Member

I cant seem to set up correctly please help, its suppose to come up as normal text!

#### MarkFL

Staff member

I cant seem to set up correctly please help, its suppose to come up as normal text!
I had it fixed, but you unfixed it... Okay, I fixed it again...dollar signs need a backslash before them, otherwise they are parsed as $\LaTeX$ tags.

#### needalgebra

##### Member

You are my hero. Thanks! #### MarkFL

Staff member

I am assuming in both cases, we want the first additional gigabyte to cost \$0.50. So, for the quadratic model, you want: $$\displaystyle f(x)=ax^2+b$$ where: $$\displaystyle f(20)=a(20)^2+b=400a+b=20$$ $$\displaystyle f(21)=a(21)^2+b=441a+b=20.5$$ I would solve by elimination. For the logarithmic model, you want: $$\displaystyle f(x)=a+b\ln(x)$$ where: $$\displaystyle f(25)=a+b\ln(25)=30$$ $$\displaystyle f(26)=a+b\ln(26)=30.5$$ I would use elimination here as well. #### needalgebra ##### Member Ok, is this right? Quadratic model: f(21) - f(20): 41a = 0.5 a = (1/2)/41 = 1/82 b = 620/41 f(x) = (1/82)x^2 + (620/41) Logarithmic model: f(26) - f(25): bLn(26) - bLn(25) = 1/2 bLn(26/25) = 1/2 b = 1/(2Ln(26/25)) = 1/Ln(26/25)^2 a + Ln(25)/Ln(26/25)^2 = 30 a = 30 + Ln(16900) f(x) = 30 + Ln(16900) + (Ln(x)/Ln(26/25)^2) f(x) = 30 + 3Ln(25) + Ln(x) #### MarkFL ##### Administrator Staff member The quadratic model is correct, but you have made an error in the logarithmic model. You have correctly found the value of$b$, but the error lies in the computation of$a$. Here is the step where I see the error: a + Ln(25)/Ln(26/25)^2 = 30 a = 30 + Ln(16900) It appears you are saying: $$\displaystyle \frac{\ln(25)}{2\ln\left(\frac{26}{25} \right)}=-\ln(16900)$$ How did you arrive at this? #### needalgebra ##### Member Honestly, I had almost no clue how to finish up the Logarithmic expression.. #### MarkFL ##### Administrator Staff member Honestly, I had almost no clue how to finish up the Logarithmic expression.. You were doing well up to the point I cited. Instead of: $$\displaystyle a=30+\ln(16900)$$ You want: $$\displaystyle a=30-\frac{\ln(25)}{2\ln\left(\frac{26}{25} \right)}$$ Do you see why? #### needalgebra ##### Member no #### MarkFL ##### Administrator Staff member You stated (in a slightly different form): $$\displaystyle a+\frac{\ln(25)}{2\ln\left(\frac{26}{25} \right)}=30$$ I have chosen to write $$\displaystyle 2\ln\left(\frac{26}{25} \right)$$ where you have used the equivalent $$\displaystyle \ln\left(\left(\frac{26}{25} \right)^2 \right)$$. So, to solve for$a$, you need to subtract $$\displaystyle \frac{\ln(25)}{2\ln\left(\frac{26}{25} \right)}$$ from both sides to obtain: $$\displaystyle a=30-\frac{\ln(25)}{2\ln\left(\frac{26}{25} \right)}$$ It's just like if you had: $$\displaystyle x+y=z$$ and you solved for$x$to get: $$\displaystyle x=z-y$$ Does this make sense? #### needalgebra ##### Member yeah i think i got it. is this correct? b=1/(2*log(26)-2*log(25)) b=12.74836584551409 #### MarkFL ##### Administrator Staff member This is closer: $$\displaystyle b\approx12.7483658455141791503275149034786729902710597570291566009838101858237$$ But...why use a decimal approximation when you can use the exact value? I would use: $$\displaystyle b=\frac{1}{2\ln\left(\frac{26}{25} \right)}$$ #### needalgebra ##### Member I've been working on several different math tasks almost non-stop for the past 10 hours or so, i have a ton of work to do that i didnt know off. All of this is due in 3 days.. so you're going to have to excuse me if i dont make sense sometimes (feeling drowsy) - besides most of this work is stuff i havent touched in a real long time. haha. what else would i have to do to the Quadratic model? how would i finish off the logarithmic model? #### MarkFL ##### Administrator Staff member You completed the quadratic model correctly, and you have the values of$a$and$b\$ for the logarithmic model as well.

$$\displaystyle f(x)=\frac{1}{82}x^2+\frac{620}{41}$$ where $$\displaystyle 20\le x$$

Logarithmic model:

$$\displaystyle f(x)=\left(30-\frac{1}{2\ln\left(\frac{26}{25} \right)} \right)+\frac{1}{2\ln\left(\frac{26}{25} \right)}\ln(x)$$ where $$\displaystyle 30\le x$$

#### needalgebra

##### Member
omg.... i feel so dumb right now. haha

Thanks for your help! Is there anyway of giving you some kind of reward on here, like givin you 500000/10 for being an amazing helper? lol if i knew you in real life i'd take you to starbucks for a cup of coffee  #### MarkFL

Staff member
Hey, if you really want to give me a reward, tell your classmates about us and encourage them to register. But, we enjoy helping here, and if you gained a bit of understanding from your time here, then our goal here at MHB has been met. I look forward to seeing you around!

#### needalgebra

##### Member
I actually study online, but i will definitely let my online - classmates know!

Staff member

#### needalgebra

##### Member
how would i graph the logarithm and quadratic models? having a little trouble here...

#### MarkFL

Staff member
Use the command:

piecewise[{{20,0<=x<=20},{x^2/82+620/41,20<x}}],piecewise[{{30,0<=x<=25},{(30-log(25)/(2log(26/25)))+1/(2log(26/25))log(x),25<x}}] where x=0 to 50

at W|A.

#### needalgebra

##### Member
i've been trying but cant seem to get it.

Staff member

#### needalgebra

##### Member
How do i get the intersection points of the logarithm and quadratic models?

#### needalgebra

##### Member
ok so for x i got :

x^2/82+620/41=
(log(26/25)*log(x))/2-log(26/25)/2+30

x=34.98726

cant seem to get y.