Polynomials- Challanging Problems ( word problems)

[wind]

Hi, i have a fre word problems that I don't know how to solve. Can someone help me with these?

1.A rectangular contaner measuring 1m by 2m by 4m is covered with a layer of lead shielding of uniform thickness. Model the volume of the shield as a function of its thickness. Find the thickness of the shielding, to the nearst centimeter, if the volume of the shielding is 3m^3.

3=(1+2x)(2+2x)(4+2x)
3=(2+2x+4x+4x^2)(4+2x)
3=(2+6x+4x^2)(4+2x)
3=(8 + 4x + 24x + 12x^2 + 16X^2 + 8X^3)
3=(8 + 28X + 28x^2 + 8x^3)
0= 8x^3 + 28X^2 + 28x + 5

a=+,- 1,5
b=+, - 1,2,4,8 a/b= +,- 1, 1/2, 1/4, 1/8, 5, 5/2, 5/4, 5/8

I tryed all of the a/b possibilites but none of them equals zero can someone check this?
afer i find that then all i need to do is to factor right?
-------------------------------------------------------------
2. A fule tank is in the shape of a horizontal right circular cylinder with a hemishpere at each end. If the lengh of the cylinder is 10m, and the volume is 2000m^3, find the common radius of the hemisheres and the cylinder.

2000= 4/3pir^3 + pir^2 (h)
2000= 4/3pir^3 + pir^2 (10)
200=4/3pir^3 + pir^2
600=4pir^3 + 3pir^2

....now what do I do?
-------------------------------------------------------------
3. Accoridnt to estimates by the Ontario Provincial Police, the stopping distance at varoius speeds is approximatley as follows.

Speed (km/h) Distance (m)
50......................22
70......................38
90.......................58
110.....................82
130....................110

if the speed, in Kilometers per hour, and y is the distance, in meters, find the polynomial y= f(x) of the least degree that fits these data.

x Y
Speed (km/h) Distance (m) 1 st difference 2nd difference
50.......................22.......................-..................-
70.......................38.....................16..................-
90.......................58.....................20..................4
110.....................82.....................24...................4
130....................110....................28...................4

2a=4
a=2

y= ax^2 + bx + c
22=2(50)^2 + b(50) +c
22= 5000 + 50b + c
c= -4978 -50b

y= ax^2 + bx + c
38= 2(70)^2 + b(70) + c
38= 9800 + 70b + c

38= 9800 + 70b -4978 - 50b
38= 4822 + 20b
-4784= 20b
-239.2=b

but in the back of the book it is 1/200X^2 + 1/5x - 1/2
I don't get it can someone expain this?

Pleas help, thanks

 

[skeeter]

first, a question ... why are you attempting to do all these problems by hand?

the polynomials in questions 1 and 2 have no rational roots, and unless you know of the "ancient" methods of solving cubic equations, then you'll need to use a calculator.

1.A rectangular contaner measuring 1m by 2m by 4m is covered with a layer of lead shielding of uniform thickness. Model the volume of the shield as a function of its thickness. Find the thickness of the shielding, to the nearst centimeter, if the volume of the shielding is 3m^3.

volume of shielding = volume of container w/ shielding - volume of container

3 = (1+2x)(2+2x)(4+2x) - (1)(2)(4)

3 = 8x^3 + 28x^2 + 28x + 8 - 8

0 = 8x^3 + 28x^2 + 28x - 3

this cubic does not have rational roots ... using a calculator, I get x = approx .097

2. 2. A fule tank is in the shape of a horizontal right circular cylinder with a hemishpere at each end. If the lengh of the cylinder is 10m, and the volume is 2000m^3, find the common radius of the hemisheres and the cylinder.

total volume = volume of cylinder + volume of 1 sphere
2000 = 10pi*r^2 + (4/3)pi*r^3
0 = (4/3)pi*r^3 + 10pi*r^2 - 2000
0 = (2/3)pi*r^3 + 5pi*r^2 - 1000

once again ... no rational roots. calculator solution is r = approx 5.96 m

3.
Speed (km/h) Distance (m)
50......................22
70......................38
90.......................58
110.....................82
130....................110

22 = a(50)^2 + b(50) + c (1)
38 = a(70)^2 + b(70) + c (2)
58 = a(90)^2 + b(90) + c (3)

subtract (1) from (2)
16 = a(70^2 - 50^2) + 20b

subtract (2) from (3)
20 = a(90^2 - 70^2) + 20b

4 = a(90^2 + 50^2 - 2*70^2)
4/(90^2 + 50^2 - 2*70^2) = a
a = .005

with a, you should be able to get b and c.

 

[wind]

^ thanks for the help!