Southwest Cleaners believes that it will need new equipment in 10 years. The equipment will cost $26,000. What lump sum should be invested today at 8% compounded semi-annually, to yield $26,000?

Let the sum be X. Solve this equation:

X*(1.04)^20 = 26,000

2.1911231 X = 26,000
X = $ 11,866.06

Thanks for your efforts and time.

Thanks for your time :-)

To find the lump sum that should be invested today at 8% compounded semi-annually to yield $26,000 in 10 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value (in this case, $26,000)
P = the principal amount (the lump sum to be invested today)
r = the annual interest rate (8% or 0.08)
n = the number of compounding periods per year (semi-annual means twice a year, so n = 2)
t = the number of years (10)

First, let's rearrange the formula to solve for P:

P = A / (1 + r/n)^(nt)

Now we can substitute the given values into the formula:

P = $26,000 / (1 + 0.08/2)^(2*10)

P = $26,000 / (1 + 0.04)^(20)

P = $26,000 / (1.04)^20

P = $26,000 / 1.83505

P ≈ $14,157.84

Therefore, approximately $14,157.84 should be invested today to yield $26,000 in 10 years at 8% compounded semi-annually.