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- A COMPANY IS CONSIDERING
EXPANDING THEIR PRODUCTION CAPACITIES WITH A NEW MACHINE
THAT COSTS $96,000
AND HAS THE PROJECTED LIFESPAN OF 8 YEARS.
THEY ESTIMATE THE INCREASED PRODUCTION WOULD PROVIDE
A CONSTANT $12,000 PER YEAR OF ADDITIONAL INCOME.
THAT MONEY CAN EARN 0.7% PER YEAR COMPOUNDED CONTINUOUSLY.
SHOULD THE COMPANY BUY THE MACHINE?
TO MAKE THIS DECISION WE'LL CALCULATE THE PRESENT VALUE
OF THIS CASH FLOW OF $12,000 PER YEAR OVER AN 8 YEARS PERIOD
THAT CAN EARN 0.7% CONTINUOUS INTEREST
AND THEN WE'LL COMPARE THAT PRESENT VALUE
TO THE COST OF THE NEW MACHINE OF $96,000.
SO THE PRESENT VALUE OF THE CASH FLOW
IS EQUAL TO THE INTEGRAL OF R OF T
WHICH IS THE ANNUAL CASH FLOW OF $12,000
x E RAISE TO THE POWER OF -KT
WHERE K WOULD BE THE CONTINUOUS INTEREST RATE
EXPRESSED AS A DECIMAL.
SO WE'D HAVE -0.007T, DT.
AND BECAUSE THE LIFESPAN IS 8 YEARS
WE'LL INTEGRATE FROM 0 TO 8.
NOTICE TO INTEGRATE THIS WE DO HAVE TO PERFORM U SUBSTITUTION.
SO WE'LL LET THE EXPONENT OF -0.007T BE EQUAL TO U.
NOTICE THAT DIFFERENTIAL U IS EQUAL TO -0.007DT.
SO IF WE SOLVE THIS FOR DT
NOTICE HOW DT IS EQUAL TO DU DIVIDED BY THIS CONSTANT.
SO LET'S GO AHEAD AND WRITE THIS IN TERMS OF U
BUT WE'LL LEAVE OFF THE LIMITS OF INTEGRATION
BECAUSE THESE ARE IN TERMS OF T NOT U.
SO WE'LL GO AHEAD AND FACTOR OUT THE 12,000.
AND THEN WRITING THIS IN TERMS OF U
WE'LL HAVE TO DIVIDE BY THIS CONSTANT HERE.
LET'S GO AHEAD AND FACTOR THAT OUT.
BUT NOW THIS WILL JUST BE E TO THE U.
IF THE INTEGRAL OF E TO THE U IS JUST E TO THE U.
SO NOW WE HAVE 12,000 DIVIDED BY -0.007
AND THE ANTIDERIVATIVE IS E TO THE POWER OF -0.007T.
AND I WILL EVALUATE THIS AT 8 AND 0
AND THEN FIND THE DIFFERENT.
LET'S GO AHEAD AND EVALUATE THIS ON THE NEXT SLIDE.
SO WHEN T IS EQUAL TO 8
WE WOULD HAVE E RAISE TO THE POWER OF -0.007 x 8.
AND THEN WHEN E IS 0 WE'D JUST HAVE E RAISE TO THE POWER OF 0
WHICH WOULD BE 1.
LET'S GO AHEAD AND EVALUATE THIS ON THE CALCULATOR.
12,000 DIVIDED BY -0.007 x E RAISE TO THE POWER OF -0.007 x 8
AND THEN - E TO THE 0 WHICH IS JUST -1 AND ENTER.
TO THE NEAREST CENT THIS WOULD BE $93,361.48.
SO THIS IS THE PRESENT VALUE OF THE EXPECTED INCREASE IN INCOME
FROM THE PURCHASE OF THE NEW MACHINE.
BUT NOTICE HOW THE MACHINE COSTS $96,000
WHICH IS MORE THAN THE PRESENT VALUE OF THE INCREASE OF INCOME
WHICH MEANS THE COMPANY SHOULD NOT PURCHASE THIS MACHINE.
WHICH MEANS THE COMPANY SHOULD NOT PURCHASE THIS MACHINE.
IF WE TAKE $96,000 - 93,000, $361.48.
LET'S GO AHEAD AND FIND THIS DIFFERENCE.
NOTICE THE DIFFERENCE IS $2,638.52.
SO WE'LL SAY THE COMPANY SHOULD NOT PURCHASE THE MACHINE
BECAUSE THE PRESENT VALUE IS $2,638.52
LESS THAN THE COST OF THE MACHINE.
I HOPE YOU FOUND THIS HELPFUL.