Homework Help! Midterm Review! College Allgebra

[vecrhite371]

Hey all,

I can't get these problems, even though I have multiple choices.

(1) You have 156 feet of fencing to enclose a rectangular region. What is the maximum area?
(a) 1517 square feet
(b) 1521 square feet
(c) 6084 square feet
(d) 24,366 square feet

I think the answer is (a), but that seems too simple (I calculated 1520.75). I feel I'm supposed to use a quadratic for this.

(2) The volume V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. A measuring device is calibrated to give V= 364in^3 when T= 260 degrees and P = 101lb/in^2. What is the volume on this device when the temperature is 300 degrees and the pressure is 251lbs/in^2?
(a) V = 168 in^3
(b) V = 188 in^3
(c) V = 148 in^3
(d) V = 12 in ^ 3

I'm pretty sure it can't be D, but every time I try to solve it none of my numbers match up to any of the answers.

(3) When the temprature stays the same, the vaolume of a gas is inversely proportional to the pressure of the gas. If a balloon is fill with 20 cubic inches of a gas at a pressure of 14 pounds per square inch, find the new pressure of the gas if the vlume is increased to 10 cubic inches?
(a) 12 pounds per square inch
(b) 5/7 pounds per square inch
(c) 26 pounds per square inch
(d) 28 pounds per square inch

I'm actually almost certain it's D, but I want to be sure.

(4) Solve the inequality
x^2-2x>=0

(a) (-∞, 0] or [2, ∞]
(b) [-2, 0]
(c) (-∞, -2] or [0, ∞)
(d) [0,2]


No idea how to do this one.


(5) Find all zeros of the function and write the polynominal as a product of linear factors.
f(x) = 3x^4 + 4x^3 + 12x^2 + 16x + 4

(a) f(x) = (3x-1)(x-1)(x+2)(x-2)

(b) f(x) = (3x+1)(x+1)(x+2i)(x-2i)
(c) f(x) = (3x-1)(x-1)(x+2i)(x-2i)
(d) f(x) = (3x+1)(x+1)(x+2)(x-2)

I have a feeling it's b, but again, I'm not sure.


(6) Use the intermediate value theorem to determine whether the polynomial function has a zero in the
given interval

f(x) = 3x^3+ 5x + 5; [-1, 0]

(a) f(-1) = 3 and f(0) = -5; yes

(b) f(-1) = -3 and f(0) = 5; yes
(c) f(-1) = 3 and f(0) = -5; no
(d) f(-1) = -3 and f(0) = -5; no

I'm pretty sure the answer is B, since the when you plug them in, that's the only answer that makes sense, but I wanna be sure.

 

[Deleted member 4993]

Hey all,

I can't get these problems, even though I have multiple choices.

(1) You have 156 feet of fencing to enclose a rectangular region. What is the maximum area?
(a) 1517 square feet
(b) 1521 square feet
(c) 6084 square feet
(d) 24,366 square feet

I think the answer is (a), but that seems too simple (I calculated 1520.75). I feel I'm supposed to use a quadratic for this.

How did you get that?

start with the fact that for a rectangle of length L and width W, the perimeter (length of fencing) is:

P = 156 = 2*(L+W) → W = 78 - L

Then

Area = A = L * W = L * (78 - L) = -L² + 78*L ← there is your quadratic equation

This is an equation of parabola - where is the vertex?

(2) The volume V of a given mass of gas varies directly as the temperature T and inversely as the pressure P. A measuring device is calibrated to give V= 364in^3 when T= 260 degrees and P = 101lb/in^2. What is the volume on this device when the temperature is 300 degrees and the pressure is 251lbs/in^2?
(a) V = 168 in^3
(b) V = 188 in^3
(c) V = 148 in^3
(d) V = 12 in ^ 3

Calculate the proportionality constant first - from the calibration numbers.

I'm pretty sure it can't be D, but every time I try to solve it none of my numbers match up to any of the answers.

(3) When the temprature stays the same, the vaolume of a gas is inversely proportional to the pressure of the gas. If a balloon is fill with 20 cubic inches of a gas at a pressure of 14 pounds per square inch, find the new pressure of the gas if the vlume is increased to 10 cubic inches?
(a) 12 pounds per square inch
(b) 5/7 pounds per square inch
(c) 26 pounds per square inch
(d) 28 pounds per square inch

I'm actually almost certain it's D, but I want to be sure.

(4) Solve the inequality
x^2-2x>=0

(a) (-∞, 0] or [2, ∞]
(b) [-2, 0]
(c) (-∞, -2] or [0, ∞)
(d) [0,2]


No idea how to do this one.


(5) Find all zeros of the function and write the polynominal as a product of linear factors.
f(x) = 3x^4 + 4x^3 + 12x^2 + 16x + 4

(a) f(x) = (3x-1)(x-1)(x+2)(x-2)

(b) f(x) = (3x+1)(x+1)(x+2i)(x-2i)
(c) f(x) = (3x-1)(x-1)(x+2i)(x-2i)
(d) f(x) = (3x+1)(x+1)(x+2)(x-2)

I have a feeling it's b, but again, I'm not sure.

Multiply the factors out to check.

(6) Use the intermediate value theorem to determine whether the polynomial function has a zero in the
given interval

f(x) = 3x^3+ 5x + 5; [-1, 0]

(a) f(-1) = 3 and f(0) = -5; yes

(b) f(-1) = -3 and f(0) = 5; yes
(c) f(-1) = 3 and f(0) = -5; no
(d) f(-1) = -3 and f(0) = -5; no

I'm pretty sure the answer is B - correct - , since the when you plug them in, that's the only answer that makes sense, but I wanna be sure.

.

 

[vecrhite371]

.

I appreciate the hints you gave me, but I honestly don't know how to do any of that. I don't know how to factor out those things with imaginary numbers, and I don't know how to calculate the proportionality, either, for the other problem.

 

[mmm4444bot]

(4) Solve the inequality
x^2-2x>=0

(a) (-∞, 0] or [2, ∞]
(b) [-2, 0]
(c) (-∞, -2] or [0, ∞)
(d) [0,2]


No idea how to do this one.

If you're unfamiliar with the algebraic method (basically, a sign chart), then perhaps a graphical method will work for you.

For what values of x is the polynomial x^2 - 2x non-negative? These will be the same values where the parabola lies on or above the x-axis.

Can you graph?

(5) Find all zeros of the function and ...

(a) f(x) = (3x-1)(x-1)(x+2)(x-2)
(b) f(x) = (3x+1)(x+1)(x+2i)(x-2i)
(c) f(x) = (3x-1)(x-1)(x+2i)(x-2i)
(d) f(x) = (3x+1)(x+1)(x+2)(x-2)

None of these answers list the zeros of f; hence, they are all invalid!

 

[mmm4444bot]

I appreciate the hints you gave me, but I honestly don't know how to do any of that.

We appreciate your honesty.

What is your situation? Are you back in school, after being away for a long time? Are you currently taking a math class?

These exercises are too advanced for a beginning-algebra student.

The volunteers here generally have little time for teaching classroom material on the boards, but we're happy to provide links to on-line lessons, if you need to re-learn the basics.

Cheers ~ Mark :cool:

 

[lookagain]

vecrhite371,

because you gave attempts and/or explanations on most of these,
I will post certain info for you.

Do you have to show your work for the problems or just mark the correct answer?

Hey all,

I can't get these problems, even though I have multiple choices.


(1) You have 156 feet of fencing to enclose a rectangular region.
What is the maximum area?

(a) 1517 square feet
(b) 1521 square feet
(c) 6084 square feet
(d) 24,366 square feet

I think the answer is (a), but that seems too simple (I calculated 1520.75).
I feel I'm supposed to use a quadratic > likely for you < for this.


In checking a solution for this, a rectangle's area is maximized when its
perimeter is in particular that of a square shape.
Divide the perimeter into fourths to get a length of a side of a square
and use that.


------------------------------------------------------------------------------

Note for the following: PV represents the product of P and V.

For this and the next problem, know that as T increases, so does V. As P
increases, V decreases. As P increases, so does T. A formula that can
express this is

(PV)/T = k, where k is a constant of proportion.


(2) The volume V of a given mass of gas varies directly as the temperature T
and inversely as the pressure P. A measuring device is calibrated to give
V= 364in^3 when T= 260 degrees and P = 101lb/in^2.
What is the volume on this device when the temperature is 300 degrees and
the pressure is 251lbs/in^2?

(a) V = 168 in^3
(b) V = 188 in^3
(c) V = 148 in^3
(d) V = 12 in ^ 3

I'm pretty sure it can't be D, but every time I try to solve it none of my
numbers match up to any of the answers.


Actually, none of the choices for this second problem are correct,
but one of the choices is within 1 cubic inch of the correct answer.


--------------------------------------------------------------------------------------


(3) When the temperature stays the same, the volume of a gas is inversely proportional
to the pressure of the gas. If a balloon is fill with 20 cubic inches of a gas at a pressure
of 14 pounds per square inch, find the new pressure of the gas if the volume is increased
to 10 cubic inches?

(a) 12 pounds per square inch
(b) 5/7 pounds per square inch
(c) 26 pounds per square inch
(d) 28 pounds per square inch

I'm actually almost certain it's D, but I want to be sure.


Because T stays the same, then the formula you can
use boils down to PV = k . . . . . (Again, it's a constant of proportion.)


--------------------------------------------------------------



(4) Solve the inequality
x^2-2x>=0

(a) (-∞, 0] or [2, ∞]
As they're presented, none of these choices are correct.
That second part must be [2, oo). Each infinity symbol always take an
(appropriate) parenthesis.


(b) [-2, 0]
(c) (-∞, -2] or [0, ∞)
(d) [0,2]


No idea how to do this one.


I am deferring this to one of the other users.

----------------------------------------------------------------------------------



(5) Find all zeros of the function and write the polynomial as a product of linear factors.

f(x) = 3x^4 + 4x^3 + 12x^2 + 16x + 4

(a) f(x) = (3x-1)(x-1)(x+2)(x-2)
(b) f(x) = (3x+1)(x+1)(x+2i)(x-2i)
(c) f(x) = (3x-1)(x-1)(x+2i)(x-2i)
(d) f(x) = (3x+1)(x+1)(x+2)(x-2)

I have a feeling it's b, but again, I'm not sure.

In this case all of the signs of the polynomial are positive.
When factored as the product of linear factors
each having only real coefficients, there can be
no factors with subtraction and/or negative
signs/symbols. (However, I am not entertaining possibilities
such as (-3x - 1)(-x - 1) or whatever.)

Right away, that fact eliminates the 1st, the 3rd, and the 4th choices.


*** Edit***: If the polynomial were 3x^4 + 4x^3 + 13x^2 + 16x + 4,
*then* one of those choices would apply.

------------------------------------------------------------------


(6) Use the intermediate value theorem to determine whether the
polynomial function has a zero in the
given interval

f(x) = 3x^3+ 5x + 5; [-1, 0]

(a) f(-1) = 3 and f(0) = -5; yes
(b) f(-1) = -3 and f(0) = 5; yes
(c) f(-1) = 3 and f(0) = -5; no
(d) f(-1) = -3 and f(0) = -5; no

I'm pretty sure the answer is B, since the when you plug them in,
that's the only answer that makes sense, but I wanna be sure.

I agree. The second answer 1) is the only choice with correct values
of f(-1) and f(0), and 2) the signs of the function values change
between those x-values.

__________________________________________________________


Another edit:

These exercises are too advanced for a beginning-algebra student.

mmm4444bot,

the student does have the words "College Algebra" in his/her headline, as well
as this was posted in the "Intermediate/Advanced Algebra" section.

 

[mmm4444bot]

the student does have the words "College Algebra" in his/her headline, as well
as this was posted in the "Intermediate/Advanced Algebra" section.

I'm suggesting that the OP is a beginner because they stated that they do not know how to do "anything" that Subhotosh suggested, including basic binomial-multiplication.

For a person in such a position, I think that these exercises are too advanced. My opinion is that this person may need to "back up".

(I have no idea whether this person has studied any algebra. They could be majoring in Sign-Holding with no math desires, other than to pass a single on-line math course required by some "school".)

Thank you for pointing out the subject line; I tend to ignore those, thus I missed the clue about a college course. My bad.

Cheers :cool:

 

[HallsofIvy]

Hey all,

I can't get these problems, even though I have multiple choices.

(1) You have 156 feet of fencing to enclose a rectangular region. What is the maximum area?
(a) 1517 square feet
(b) 1521 square feet
(c) 6084 square feet
(d) 24,366 square feet

I think the answer is (a), but that seems too simple (I calculated 1520.75). I feel I'm supposed to use a quadratic for this.

You don't say how you calculated that, but "1520.75" is not correct. You know that, taking "L" as the length of the rectangle in feet and "W" is the width the rectangle in feet, 2L+ 2W= 156 so that W= 78- L. The area, then, is LW= L(78- L)= 78L- L^2. One way to find the maximum possible value of that is to complete the square. Can you do that?

(2)The volume V of a given mass of gas varies directly as the temperature T
and inversely as the pressure P. A measuring device is calibrated to give V= 364in^3 when T= 260 degrees and P = 101lb/in^2. What is the volume on this device when the temperature is 300 degrees and the pressure is 251lbs/in^2?
(a) V = 168 in^3
(b) V = 188 in^3
(c) V = 148 in^3
(d) V = 12 in ^ 3

I'm pretty sure it can't be D, but every time I try to solve it none of my numbers match up to any of the answers.

Again, you don't say how you did that but, no, (d) is NOT correct.
" The volume V of a given mass of gas varies directly as the temperature T
and inversely as the pressure P" means that V= KT/P for some number K. You can use the first given values of P, T, and V to find K and then do the arithmetic.
Another way to do this is to find the ratios of the given temperatures and pressures: 300/260 and 251/101. Since V "varies directly as the temperature", multiply the first volume by 300/260 and since it "varies inversely as the pressure", divide that by 251/101.

(3) When the temprature stays the same, the vaolume of a gas is inversely proportional to the pressure of the gas. If a balloon is fill with 20 cubic inches of a gas at a pressure of 14 pounds per square inch, find the new pressure of the gas if the vlume is increased to 10 cubic inches?
(a) 12 pounds per square inch
(b) 5/7 pounds per square inch
(c) 26 pounds per square inch
(d) 28 pounds per square inch

I'm actually almost certain it's D, but I want to be sure.

First, is this really what the problem says? It's impossible for volume to increase from 20 to 10!
Did you mean "decrease" from 20 to 10? Or increase from 10 to 20? Or increase by 10 cubic inches?

Again, saying "the volume of a gas is inversely proportional to the pressure of the gas" means that V= K/P for some number K. You can set V= 20 and P= 14 and solve for K. Then (assuming the the volume decreases[/b ] from 20 to 10), solve 10= K/P, with that value of K, for P. Or, more simply, note that in going from 20 to 10 the volume is multiplied by 1/2 so, since this is an inverse proportion, the volume with be multiplied by 2.

(4) Solve the inequality
x^2-2x>=0

(a) (-∞, 0] or [2, ∞]
(b) [-2, 0]
(c) (-∞, -2] or [0, ∞)
(d) [0,2]


No idea how to do this one.

x^2- 2x= x(x- 2)>= 0. A product of two numbers is positive if and only if they have the same sign- both positive or both negative. What values of x satisfy both x> 0 and x-2> 0? What values of x satisfy both x< 0 and x-2< 0?

(5) Find all zeros of the function and write the polynominal as a product of linear factors.
f(x) = 3x^4 + 4x^3 + 12x^2 + 16x + 4

(a) f(x) = (3x-1)(x-1)(x+2)(x-2)

(b) f(x) = (3x+1)(x+1)(x+2i)(x-2i)
(c) f(x) = (3x-1)(x-1)(x+2i)(x-2i)
(d) f(x) = (3x+1)(x+1)(x+2)(x-2)

I have a feeling it's b, but again, I'm not sure.

A sum of products of positive numbers cannot be 0 so there must be some negative terms so that there are no positive zeros. That means that (a) (because of the "3x-1" and "x-1" which say that 1/3 and 1 are zeros), (c) (because of the "3x-1"), and (d) (because of the "x-2") cannot be correct. That leaves only (b) but I am concerned about your "feeling" without knowing why.

(6) Use the intermediate value theorem to determine whether the polynomial function has a zero in the
given interval

f(x) = 3x^3+ 5x + 5; [-1, 0]

(a) f(-1) = 3 and f(0) = -5; yes

(b) f(-1) = -3 and f(0) = 5; yes
(c) f(-1) = 3 and f(0) = -5; no
(d) f(-1) = -3 and f(0) = -5; no

I'm pretty sure the answer is B, since the when you plug them in, that's the only answer that makes sense, but I wanna be sure.

That is frankly a silly problem. Yes, f(0)= 5 so (b) is the only one that can be correct. But suppose the problem had NOT given you the formula for the f(x) but only said
"(a) f(-1)= 3 and f(0)= -5
(b) f(-1)= -3 and f(0)= 5
(c) f(-1)= 3 and f(0)= -5
(d) f(-1)= -3 and f(0)= -5"
would you have been able to determine which had a zero in [-1, 0]? Do you know what the "intermediate value theorem" says?

Frankly, I am concerned about your repeatedly saying "I'm pretty sure" or "I have a feeling" without giving reasons! You don't solve math problems solely on "feelings". You need to know what you are doing.

 

[Deleted member 4993]

Frankly, I am concerned about your repeatedly saying "I'm pretty sure" or "I have a feeling" without giving reasons! You don't solve math problems solely on "feelings". You need to know what you are doing.

That's the problem with multiple choice questions!

I would blame the teacher - s/he is probably trying to find a way out of correcting exam papers - probably on-line course. All the on-line courses should administer proctored exams!!

 

[mmm4444bot]

All the on-line courses should administer proctored exams!

If implemented in the United States, this idea may put math students at risk of failing their on-line math course.

The University of Phoenix could end up having to close several office buildings around the country.

Future scientists may lose their fear of equations.

And, in the long term, math-and-science rankings for our country may move above 17th place on the planet.

Gosh! What were you thinking!!