Using the time value of money Helen wants to take the next four years off work to travel around the world. She estimates her annual cash needs at $31,000 (if she needs more, she will work odd jobs). Helen believes she can invest her savings at 10% until she depletes her funds. Requirements 1. How much money does Helen need now to fund her travels?

Helen needs $143871 to fund her travels.

Step-by-step explanation:

This is a present value annuity as Helen wants to save money in order to withdraw $31000 per annum therefore that is the payment she will get per annum for four straight years for her holiday which will sustain her without working during holidays.

Given :

Payment that she will be withdrawing on her savings per annum for holidays $31000

R interest rate of the savings which is 10%

N the number of periods for her withdrawals which is 4 years  

Formula used is the present value annuity formula which is

Present value annuity= payment ((1-(1/ ((1+R) ^N)/R))*(1+R) ^N

                                        =$31000 ((1-(1/((1+0.1)^4)/0.1))*(1+0.1)^4

                                         =$143871

Therefore Helen must save $143871 at a 10% saving rate in order to withdraw $31000 per annual continuously for four years consecutively on holiday in order to fulfill her requirements of not working for the whole four years.  

We substitute with the given values in order to get how Helen will be able to save for the whole four years because we already know the payment that is going to be made each year to her for holiday is $31000 which will satisfy her needs.  so Helen needs $143871

a.   we have two

lines:   y = 4-x   and  

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b.   4-x = 8-x^-1

  x           4-x       8-x^-1

-3           7        

8.33

-2           6         8.5

-1           5         9

  0           4        

-

  1           3        

7

  2           2        

7.5

  3           1         7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:  

x=-4.236

y = 4-x   = 4 – (-4.236)

= 8.236

y = 8-x^-1 =   .236)^-1

= 8.236

x=0.236

  y = 4-x   = 4 – (0.236) = 3.764

y = 8-x^-1 =   8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c.   to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x   and   y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines.   the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!