What are the answers to these math questions? Mostly Algebra 1&2?

Question

I'd appreciate if you could tell me how you got the answer... but otherwise it is okay...
1.) Give all possible zeros for the following polynomial.
P(x)= 2x^3+8x^2+12x-8

2.) Give the domain and range for the rational function.
f(x)= 1/(x+8)

3.) A rectangular piece of cardboard measuring 26 inches by 47 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. For what value of x will the volume be a maximum? If necessary, round to 2 decimal places.

4.) The volume of V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. If V=176.0 in^3 when T= 220° and P=15 Ib/in^2, what is the volue when T=150° and P=20 Ib/in^2?

5.) If an object is propelled upward from a height of 160 feet at an initial velocity of 144 feet per second, then its height after t seconds is given by the equation h(t)= -16t^2+144t+160 After how many seconds does the object hit the ground?

Thankss :) That's all! :D

Answer 1

I have a method for finding out roots with Excel that I would be happy to share with you. There is only one root and it lies between 0.4 and 0.5. To find the root, I would use Newton's method:



x1 = x0 - f(x0) / f'(x0)



f'(x) = 6x^2 + 16x + 12



Within three iterations, you get 0.488302...



***



The domain is all real numbers such that x <> -8. If x = -8, the denominator is zero, which is not permitted.



The range is all real numbers such that y <> 0. When x > -7, the denominator gets larger as x increases, resulting in a fraction that gets closer and closer to 0 but never reaches it. When -8 < x < -7, the denominator is a positive fraction, so the value of f(x) is > 1. In the regions where x < -9 and -9 < x < -8, the same things are true EXCEPT that the results is a negative number, so that as x decreases, f(x) is also asymptotic to y = 0, but from the negative side.



***



Since you cut x out of each end, the box measures (26 - 2x) * (47 - 2x) on its base, and the height is x.



V(x) = (26 - 2x) * (47 - 2x) * x

V(x) = 1222x - 146x^2 + 4x^3



V'(x) = 1222 - 292x + 12x^2

12x^2 - 292x + 1222 = 0



x = (-(-292) +/- sqrt((-292)^2 - 4 * 12 * 1222)) / (2 * 12)

x = (292 +/- sqrt(85264 - 58656)) / 24

x = (292 +/- sqrt(26608)) / 24

x = (292 +/- 163.11) / 24

x = 455.12/12 or x = 128.88/12

x = 10.74 or x = 37.92



Now, obviously x <> 37.92, because 26 - 2 * 37.92 < 0, so x = 10.74



***



V = k * T/P

176.0 in^3 = k * 220º / 15 lb/in^2

k = 176.0 in^3 * 15 lb/in^2 / 220º

k = 12 lb-in/º



V = 12 lb-in/º * 150º / 20 lb/in^2

V = 90 in^3



***



h(t) = -16t^2 + 144t + 160

-16t^2 + 144t + 160 = 0

t^2 - 9t - 10 = 0

(t - 9)(t + 1) = 0

t = 9 or t = -1



Obviously t = -1 is not permitted, so t = 9 seconds.

Answer 2

I'll answer the first 2 questions for you.



1. Finding zeroes can be quite tricky, but there are some rules in limiting the possibilities.

Firstly, simplify the equation...



P(x) = 2(x^3 + 4x^2 + 6x - 4) You only need to look at the part in parenthesis.



A rule is that the zeroes of a polynomial is a factor of the constant (-4) divided be a factor of the leading term (1, which is the coefficient of x^3)



This means that the possible zeroes are 4, -4, 2, -2, 1, and -1.



You can check them yourself (learn synthetic division if you haven't already),but the answer is

P(x) = 2(x - 2)(x^2 + 2x - 2)



If you need to, use the quadratic formula to find the roots for x^2 + 2x - 2 (answer is -1 ± √3)



2. A fraction cannot have a denominator of 0, so the domain is any real number except for when

(x + 8) = 0, or when x = -8

The range can be any real number

Answer 3

1) 2x^3+8x^2+12x-8=0 when x=0.488302



http://www.wolframalpha.com/input/?i=2x%5E3%2B8x%5E2%2B12x-8%3D0



2)Domain: For all real values of x except 8 bcoz if x=8 then it will become undefined.

Range: the function will have all possible values except 0. Mathematical expressions in the link.



http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=domain%20and%20range%20of%20f(x)%3D%201%2F(x%2B8)



3)you can utilize only the shorter side fully. So you will have a square cardboard with side 26 inches to make the box. so the size of the side of the square=26/3=8.67



4)V=kT/P

So, putting the given values we will get k=(176*15)/(220-273)=-50 (app.)

So, put the value of k in the new equation to get v

i.e. V=(-50)*(150-273)/20 =307.5 in^3



(i subtracted 273 because the temp. you gave is in degree celsius. so you will have to convert that into kelvin scale to calculate)



5) for getting the time, you will need the ht. travelled by the body. And to get the distance propelled you will need projectile formula where gravity will play a role and you will need the mass. Otherwise, it can be a tricky question, so i will have to think a little bit more. if i get the answer i will edit my answer.



Well, thats all i could do. i can be wrong but i gave my best to solve the problems.(and all for just 10 points:))



..........And a Happy New Year!



EDIT: the object is propelled upwards. so it can go up to any ht. which is unknown. So technically, the question is wrong. and tychabrahe has solved this one without the help of the data "velocity of 144 feet per second". So something's wrong in it. Moreover tychabrahe has taken ht. as 0, whereas the reference level i.e pt. of projection is 160ft high from ground.

Answer 4

1.) commission, C = 2(14,000x) / 100 C = 28,000x / one hundred C = 280x ¯¯¯¯¯¯¯¯ 2.) Convert the equation to slope-intercept form: x + 3y = 12 3y = - x + 12 y = - 1/three x + four Slope = - 1/three A perpendicular line can have a slope that's the bad reciprocal of the slope of the perpendicular line, in this case, three. Common equation of perpendicular line: y = 3x + b because you didn't provide a factor that the line goes by way of, b can equal some thing.