Cara wants to attend a two-year community college that will cost $23,000 a year. By working during the summer, she can save $2,800 each year for four years. She has identified $10,000 in scholarship money that she would qualify for. Her family is starting a college fund for her by depositing money in an account that will yield 8% compound interest, compounded quarterly for four years. To the nearest thousand, how much must her parents deposit so that their contribution will make up the shortfall in Cara’s plans?(1 point)
Cara's total cost for attending the two-year community college is $23,000 x 2 = $<<23000*2=46000>>46,000.
Her total savings from working in summer are $2,800 x 4 = $<<2800*4=11200>>11,200.
So, her total expenses that are covered by savings and scholarships are $11,200 + $10,000 = $<<11200+10000=21200>>21,200.
Her total expenses after savings and scholarships are deducted are $46,000 - $21,200 = $<<46000-21200=24800>>24,800.
The compounding formula is given by A = P (1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.
The equation to find the principal amount, P, is P = A / ((1 + r/n)^(nt)).
Using the given values, the principal amount, P, is P = $24,800 / ((1 + 0.08 / 4)^(4 x 4)).
Therefore, the principal amount that must be deposited is P = $<<24800/((1+0.08/4)^(4*4))=19754>>19,754. Answer: \boxed{19754}.