Mika

A 150.0 kg cart rides down a set of tracks on four solid steel wheels, each with radius 20.0 cm and mass 45.0 kg. The tracks slope downward at an angle of 15 ∘ to the horizontal. If the cart is released from rest a distance of 13.5 m from the bottom of the track (measured along the slope), how fast will it be moving when it reaches the bottom? Assume that the wheels roll without slipping, and that there is no energy loss due to friction.

I've been stuck on this problem for a while now. I've calculated the kinetic energy for the cart and the 4 wheels to solve for the velocity. However, my answers are wrong. I've gotten 6.2,6.9, 8.3 but none work. Any ideas would help!

The combination of a clockwise rotation about (0,0) by 120 followed by a clockwise rotation about (4,0) by 60 is a rotation. Find the coordinates of its center and angle of rotation.

The two rotations are a products of reflections however how would I find the center. My teacher said it is (1, sqrt(3)) and the angle of rotation is 180.

I know how to find the angle of rotation but not the center, if anyone can help. that would be great.

Identify the combination formed by first translating by the vector (2,0) and then rotating by 90◦ about (0,0).

My teacher said it is a rotation centered at (−1,1) by an angle of 90◦. Can someone help find out how my teacher got this answer?

In a right angled spherical triangle α = a which both does not equal 90◦. Find b and c.

Hi guys, I'm trying to do some practice problems and I'm stuck with this problem.

So lets say that the angle opposite from b is β and the angle opposite from c is γ. Since it is a right angle spherical triangle means either β or γ is 90. That seems to be the only information I am able to extract from this question.

I know about Napier's rule for spherical right triangles but I only have one information. Any help would be appreciated.

Question: dy/dt+ty=1+t where y(3/2)=0

So i know this is a non-homogeneous equation and as I started working on it, I ended up with:

e^t2/2 × y=∫t3/2 e^t2 dt

and I'm stuck at this step. Any help?

Can you simplify this? I have ideas but I'm not sure what could be considered illegal or not.

c=(a^2-a^3-ab^2)/(a^2+b^2)

So i pulled an a from the numerator:

c=a(a-a^2-b^2)/(a^2+b^2)

I'm not sure but i think i can pull an a^2+b^2 in the numerator.

Any ideas? Thanks!

Problem: Prove 7n+6 <= 3^n for all n>2 in N using induction.

I'm having trouble with the algebra part.

so for:

base case: n=3 7(3)+6<=3^3 which holds.

next step:

we assume for k in N where k>2 that 7k+6<=3^k then we must prove:

7(k+1)+6<=3^(k+1)

so i started of like this:

7(k+1)+6=(7k+6)+7<=3^k+7

now i'm having trouble showing this is less

than 3^(k+1)

anyone wanna help? thanks.

Hi all ! I have an exam coming up and I'm looking at congruence class proofs and I stumbled upon this one and I'm not sure how to prove it. Any proofs or ideas?

QUESTION:

In Zn (n> or equal to 2) [a]n is invertible if and only if gcd(a,n)=1

Let P2 have the inner product:

<P,Q> integral from 1 to 0 p(x) q(x) dx

Apply the Gram–Schmidt process to transform the standard basis S={1,x,x^2}

into an orthonormal basis.

I know how to do the Gram–Schmidt process on regular inner products with number but I can't seem to get it when it involves integral.

Anyone want to help?

Question:

Find an inverse of a modulo m for each of these pairs of relatively prime integers using euclidean algorithm and backtracking.

a=55 and m=89

So I did gcd(89,55) and worked backwards and I got my inverse to be 60 but when I test it, it is not divisible by 89. So i did something wrong.

My work:

gcd(89,55)

89=1*55+34

55=1*34+21

34=1*21+13

21=1*13+8

8=1*5+3

5=1*3+2

3=1*2+1

2=1*2

now working backwards:

1=3-1*2

1=3-1*(5-1*3)

1=-1*5+4*3

1=-1*5+4*(8-1*5)

1=4*8+-5*5

1=4*8+-5*(13-1*8)

1=-5*13+9*8

1=-5*13+9*(21-1*13)

1=9*21+-14*13

1=9*21+-14*(34-1*21)

1=-14*34+23*21

1=-14*34+23*(55-1*34)

1=23*55+-37*34

1=23*55+-37*(89-1*55)

1=-37*89+60*55

i got my inverse as 60 but the book says 34

Anyone wanna help?

Hi guys! I have a test soon so I've been trying to do some problems and i'm stuck on this one.

QUESTION:

Let (u,v) be the inner product on R^2 generated by [2 1, 1 1] and let u=(2,1), v=(-1,1), w=(0,-1). Compute:

d(v,w), I know d means distance but im not sure how to do this.

Suppose that the characteristic polynomial of some matrix A is found to be p(x)=(x-1)(x-3)^2(x-4)^3

a. What is the size of A?

So the eigenvalues are x=1, 3,4

So since there are 3 distinct values, does that mean the size is 3x3

b. Is A invertible?

Yes A is invertible since none of the eigenvalues are 0.

c. How many eigenspaces does A have?

I'm not sure about this one but since there are needs to be independent eigenvector, would it be 3.

Any help would be appreciated.

QUESTIONS:

1. If A is a 3x5 matrix, then the rank of A is at most ___. Why?

2. If A is a 3x5 matrix, then the nullity of A is at most ___. Why?

3.If A is a 3x5 matrix, then the rank of A^T is at most ___. Why?

4. If A is a 3x5 matrix, then the nullity of A^T is at most ___. Why?

MY ANSWER:

1. 3

2. 2

3. 3

4. 3

I'm not sure if 4 is 3 or 2?

anyone can verify my answers? THANKS!

Show that if A and B are sets |A|=|B|, then |P(A)| =|P(B)|

So, do i have to have two cases with A being uncountable, countable. Any help would be appreciated!

If A has size nxn and In,A,A^2,...,A^n^2 are distinct matrices, then {In, A, A^2,...,A^n^2} is linearly dependent.

The answer key says this is true but i'm not seeing it. Anyone wanna help?

The set of all nxn matrices A such that AB=BA for some fixed nxn matrix B

So, i'm trying to work this out:

I know the zero vector is in there

but how do i do the closure under addition and multiplication.

Any insights? Thanks.

Hi guys! I'm studying for an exam and I'm looking at this true or false question:

QUESTION: The linear combinations a1v1+a2v2 and b1v1+b2v2 can only be equal if a1=b1 and a2=b2.

The answer key says its false but I'm not understanding why.

So i know a linear combination by definition is for some vector w in a vector space v, then w is a linear combination of vectors v1,v2...vr in V if w can be expressed as:

w=k1v1+k2v2+...+krvr where k1,k2..kr are scalars

So does this mean, in order for them to be equal:

a1v1+a2v2 and b1v1+b2v2

means:

a1v1+a2v2=k1w1+..+krwr where w is a vector in the same vector space as v and k1..kr are constants

which means b1v1+b2v2 can only be equal if it contains those vectors?

or would it be a1=-b1 and a2=-b2

since we can isolate them:

a1v1=-a2v2.

I dont know, any help would be appreciated!

Show that if A and B are finite sets, then A ∪ B is a finite set.

Can someone help me with this proof?

So i know i have to prove:

1. A intersect (B-A) is a subset of the empty set

2. The empty set is a subset of A intersect (B-A)

I've got the idea for 1 but I need help on 2.

Anyone wanna help me start it off? thanks!

QUESTION:

The set of all pairs of real numbers of the form (1,x) with the operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky)

So the answer key says yes but I'm not understanding how to show it

If anyone can help me with the three basic axioms that would be great. For example:

Closure under addition, multiplication and showing the zero vector.