Homework Help
Q.
Write the function in terms of unit step functions, then find the laplace transform of the given function.
f(t)= 4 if 0<=t<3
f(t)= -2 if t>=3
f(t)= 4 if 0<=t<3
f(t)= -2 if t>=3
Ans.
Sunday 25 May 2014
Second order equations: reduction of order
Homework Help
Q.
\(y_{1}(t)=e^{-3t}, y''+6y'+9y=0\)
Find the general solution and a second linearly independent solution of y2, where 
using reduction of order method.
Ans.
Definitions of Laplace Transform
Homework Help
Q.
Find the Laplace Transform of the function f(t).
(e^(-2t)-1)^2
Ans.
Series Geometry Question - (Please help and you will be a Lifesaver!)
Homework Help
Q.
Consider a square inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having a width of 1. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles
Ans.
ARC LENGTH
Homework Help
Q. Consider the curve r=(e^(-4t)cos(-7t), e^(-4t)sin(-7t), e^(-4t). compute the arclength fuction s(t): (with initial point t=0).
Ans.
r'(t) = (e^(3t) (3 cos(-4t) + 4 sin(-4t)), e^(3t) (3 sin(-4t) - 4cos(-4t)), 3e^(3t))
So, ||r'(t)|| =
= e^(3t) * sqrt[(3 cos(-4t) + 4 sin(-4t))^2 + (3 sin(-4t) - 4cos(-4t))^2 + 3^2]
= e^(3t) * sqrt[9 + 16 + 9], via cos^2(x) + sin^2(x) = 1
= sqrt(34) * e^(3t).
= e^(3t) * sqrt[(3 cos(-4t) + 4 sin(-4t))^2 + (3 sin(-4t) - 4cos(-4t))^2 + 3^2]
= e^(3t) * sqrt[9 + 16 + 9], via cos^2(x) + sin^2(x) = 1
= sqrt(34) * e^(3t).
Thus, the arc length function s(t) equals
integral(0 to t) sqrt(34) * e^(3t) dt
= (sqrt(34)/3) * (e^(3t) - 1)
Ref: Image source
integral(0 to t) sqrt(34) * e^(3t) dt
= (sqrt(34)/3) * (e^(3t) - 1)
Ref: Image source
Business math
Homework Help
Q. Adam wants to own a house in 5 years. In order to do so he takes his $20,000 and deposit it in an account that pays 9% compounded quarterly, and leave it there for five years. After 5 years, Adam take out $30,000, leaving the rest in the account for 25 more years. Adam takes the 30,000 dollars and he make a down payment on a house that cost $200,000 Adam finances the other $170,000 at 6% compounded monthly for 30 years. At the end of 10 years, he refinance the house (paying no fees) for 10 years at 5% compounded monthly. Because of the refinance, they save a certain amount of money on their payment every month. They take this extra money every month, and deposit it an annuity which pays an annual rate of 14.5% compounded monthly. At the end of the 15 years, their house is paid off and they have accumulated a lot of money in two accounts the original leftover money which has been accruing interest for 25 years, and the annuity which has had monthly investments for 15 years. How much money does he have in total? How much did he pay for the house?
Ans.
Phase 1) He invests $20,000 for 5 years at 9% returns quarterly.
Hence amount accumulated at 5 years = $ 20,000*(1+9%/4)^(4*5) = $ 31,210.18401
He withdraws $ 30,000 out of this. Hence residual in account which will still get compounded = $1,210.18401
This stays for another 20 years => value of this residue after 25 more years = $1,210.18401 *(1+9%/4)^(4*25) = $ 11,205.17034 (A)
Phase 2) Downpayment of $ 30,000 (B) and Loan of $ 170,000 for 30 years compounded monthly.
=> Monthly interest = $ 170,000 * (6/12)% = $ 850
Total interest paid = $ 850*12*10 = $ 102,000 (C)
Phase 3) Converts loan to rate 5% compounded monthly for 10 years
=> New interest = $ 170,000 * (5/12)% = $ 708.33333
Total interest paid = $ $ 708.33333*12*10 = $ 85,000 (D)
And Principle paid = $ 170,000 (E)
Hence interest saved = $ 850 - $ 708.3333 = $ 141.6666667
Phase 4)
Equal deposit = $ 141.6666667 per month.
=> After 15 years, this will accrue a total of Future woth of annuity at 14.5% compounded monthly = $ 141.6666667 * (1+(1+14.5%/12)^1+(1+14.5%/12)^2++(1+14.5%/12)^3+...+(1+14.5%/12)^(12*15-1) = $ 90137.36411 (F)
Hence total money he has = A+F = $ 101,342.5345 = ANswer
Hence total money he paid = B+C+D+E = $ 387,000 = Answer
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