Suppose that Ramos contributes $5500/year into a traditional IRA earning interest at the rate of 3%/year compounded annually, every year after age 37 until his retirement at age 67. At the same time, his wife Vanessa deposits $3850/year (the amount after paying taxes at the rate of 30%) into a Roth IRA earning interest at the same rate as that of Ramos. Suppose that Ramos withdraws his investment upon retirement at age 67 and that his investment is then taxed at 30%

A) How much will Ramo's investment be worth (after taxes) at that time?
B) How much will Vanessa 's investment be worth at that time?

(a)$9344.96

(b)$9344.96

Step-by-step explanation:

(a)Ramos Contributes $5500/year

Therefore his principal =$5500

Rate, r=3%

Time = 67-37 =30 years

Since it is compounded annually, Period =1

Ramos Tax at 30% = 30% of 13349.94 =$4004.98

(b)Vanessa Contributes $3850/year

Therefore her principal =$3850

Rate, r=3%

Time = 67-37 =30 years

Since it is compounded annually, Period =1

Ramos Savings after taxes$9344.96

Vanessa Savings$9344.96

Step-by-step explanation:

Ramos Savings:        

RateInterest rate5%        

NperNumber of years30(67-37)      

Pmt         Amount of deposit per year$5,500      

FV Future value of savings at age 67$365,413.66(using FV function of excel with Rate=3%,Nper=30, Pmt=-5500)  

Excel Command: FV(3%,30,-5500)        

Taxes on savings=30%*365413.66=$109,624.10      

Ramos Savings after taxes$9344.96    

Vanessa Savings:        

RateInterest rate3%        

NperNumber of years30(67-37)      

PmtAmount of deposit per year$3,850        

FVFuture value of savings at age 67$255,789.56(using FV function of excel with Rate=3%,Nper=30, Pmt=-3850)  

Excel Command: FV(3%,30,-3850)        

Vanessa Savings$9344.96        

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.