Difficulty understanding a financial question

Question

To have $\$50,000$ for college tuition in $20$ years, what gift $y_o$ should a grandparent make now? Assume $c = 10\%$. What continuous deposit should a parent make during $20$ years? If the parent saves $s = \$1000$ per year, when does he or she reach $\$50,000$ and retire?

I think the gift a grandparent should make is $y_0=50000e^{-0.1\cdot20}=50000/e^2\approx6766.76$ dollars. But then I do not understand the continuous deposit that a parent should make. Is the grandparent's gift not enough to have $\$50,000$ in $20$ years?

Answer 1

1. $y_0$ is the present value of $y=\$\, 50,000$, that is $$ y_0=y\,\mathrm e^{-ct}=50,000 \times \mathrm e^{-0.1 \times 20}\approx \$\, 6,766.76 $$
2. starting with $y_0=0$ the continuous deposit $s$ to obtain $y$ in $t=20$ years is found by $$ y=\frac{s}{c}\left(e^{ct}-1\right) $$ that is $$ s=\frac{yc}{e^{ct}-1}=\frac{50,000\times 0.1}{e^{0.1\times 20}-1}=\frac{5,000}{e^{2}-1}\approx \$\, 782.59 $$
3. with a continuous deposit $s=\$\,1,000$ we have $$ y=\frac{s}{c}\left(e^{ct}-1\right)\quad\Longrightarrow\quad t=\frac{1}{c}\log\left(\frac{yc}{s}+1\right) $$ that is $$ t=\frac{1}{0.1}\log\left(\frac{5000}{1000}+1\right)=10\log 6\approx 18 \text{ years} $$