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1)Grandmother Helena sets up a trust fund for her granddaughter Emma. Helena's fool-proof investments on the money in the trust fund make interest at a rate of I, measured in units of year^1. She removes money from the trust fund to give to her granddaughter at a rate of q(t)dollars per year.

Let x(t), in dollars, be the amount in the trust fund at t years after the inception of the fund. Find x in terms of x, q and the interest rate I.

x'=??

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2)Increasing allowance

1 point possible (graded)

Grandmother decides to give a monthly allowance to her granddaughter from the trust fund. The allowance is to start at $ 0 at the birth of her granddaughter, and then increase by $ 10 each month afterwards.

Describe the grandmother's gifts, approximately, in terms of a continuous rate q(t) (in dollars per year).

(Use a linearly increasing function with no constant term for q(t).)

q(t)=???

3)Solve the differential equation

Assume that the interest rate stays constant at I (year^1) and that Grandmother Helena carries out her plan of steadily increasing the monthly allowance to her granddaughter as described above. Find the general solution to the DE that models the amount of the trust fund.

(In other words, solve the DE you wrote down in the first part of this question with the specific q(t) that you found above. Your answer should be in terms of t and I only. Use C for the parameter in the general solution so that it is the coefficient of a homogeneous solution)

(Simplify your expression for C as much as possible; e.g. instead of C+1 type C. Instead of C10 or C/10 type C.)

x(t)=?????

4)Let the fund run dry

1 point possible (graded)

Grandmother plans to pay the allowance to her granddaughter as described before but also wants the trust fund to decrease to 0 when she becomes an adult at 18 years of age. If the interest rate is constant at 1% per year, find the amount x(0) that she should put in the trust fund when her granddaughter is born. (Enter your answer to the nearest thousand dollars.)

x(0)=?????

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