Question

Because of a chronic water shortage in California, new athletic fields must use artificial turf or xeriscape landscaping. If the value of the water saved each quarter is $3,500, how much can a private developer afford to spend now on artificial turf provided he must recover his investment in 5 years. Use an interest rate of 9% per year, compounded continuously?

Answer 1

To calculate the amount a private developer can spend on artificial turf, we use the concept of present value. Given an interest rate of 9% per year, compounded continuously, we determine the present value of the future cash flows from water savings. The amount a private developer can afford to spend is approximately $1,511.06.

Step-by-step explanation:

To calculate how much a private developer can afford to spend on artificial turf, we will use the concept of present value. Given an interest rate of 9% per year, compounded continuously, we need to determine the present value of the future cash flows generated by the water saved each quarter.

Using the formula for continuous compounding, we can calculate the present value:

Present Value = Future Value / e^(interest rate * time)

where e is Euler's number (approximately 2.71828), and time is the number of quarters (5 years = 20 quarters).

Plugging in the values, the present value of the water saved is:

Present Value = $3,500 / e^(0.09 * 20)

Calculating this value, a private developer can afford to spend approximately $1,511.06 on artificial turf now.