Final answer:
To calculate the amount James will deposit in an annuity account for his child's college fund to reach $60,000 in 12 years at an APR of 3.3% compounded quarterly, we use the future value annuity formula. We solve for the quarterly deposit amount (PMT) and multiply it by the total number of deposits to find out how much James will deposit. The interest earned is the difference between the future value and total deposits.
Step-by-step explanation:
Jude's family is budgeting for a road trip and has $600 to spend on accommodations. To describe this situation with a linear inequality where x represents the number of nights camping and y represents the number of nights in a hotel, we start by writing an equation that reflects the total cost of both choices not exceeding the budget of $600. The equation for the cost of camping is $20 per night, and the cost for a hotel room is $140 per night.
The linear inequality that represents this situation is:
20x + 140y ≤ 600
To solve the mathematical problem completely, Jude's family can plug in different values for x and y to see the combination of nights camping and staying in a hotel that will keep them at or under budget. For example, if they choose to camp for 10 nights (x = 10), they could calculate how many nights they could afford to stay in a hotel by rearranging the inequality to solve for y:
20(10) + 140y ≤ 600
200 + 140y ≤ 600
140y ≤ 400
y ≤ 2.857
Since they cannot stay in a hotel for a fraction of a night, the maximum number of hotel nights is 2 when they camp for 10 nights.
Formula:
60000 = PMT * (((1 + 0.033/4)^(4*12) - 1) / (0.033/4))
We calculate this equation to find the value of PMT, which is the quarterly deposit amount.
After finding PMT, we multiply it by the total number of deposits (48) to find out how much of the $60,000 James will ultimately deposit.
The interest earned is then calculated by subtracting the total deposits from the future value of $60,000.