The model t = 16.625 In (xx - 750), x > 750 approximates the length of a home mortgage of 50,000

The model t = 16.625 In (xx - 750), x > 750 approximates the...

The model $ t = 16.625 In \left(\dfrac{x}{x - 750}\right), x > 750 $ approximates the length of a home mortgage of $ \$150,000 $ at $ 6\% $ in terms of the monthly payment. In the model, $ t $ is the length of the mortgage in years and $ x $ is the monthly payment in dollars. (a) Use the model to approximate the lengths of a $ \$150,000 $ mortgage at $ 6\% $ when the monthly payment is $ \$897.72 $ and when the monthly payment is $ \$1659.24 $. (b) Approximate the total amounts paid over the term of the mortgage with a monthly payment of $ \$897.72 $ and with a monthly payment of $ \$1659.24 $. (c) Approximate the total interest charges for a monthly payment of $ \$897.72 $ and for a monthly payment of $ \$1659.24 $. (d) What is the vertical asymptote for the model?Interpret its meaning in the context of the problem.

The model

$ t = 16.625 In \left(\dfrac{x}{x - 750}\right), x > 750 $

approximates the length of a home mortgage of $ \$150,000 $ at $ 6\% $ in terms of the monthly payment. In the model, $ t $ is the length of the mortgage in years and $ x $ is the monthly payment in dollars.

(a) Use the model to approximate the lengths of a $ \$150,000 $ mortgage at $ 6\% $ when the monthly payment is $ \$897.72 $ and when the monthly payment is $ \$1659.24 $.

(b) Approximate the total amounts paid over the term of the mortgage with a monthly payment of $ \$897.72 $ and with a monthly payment of $ \$1659.24 $.

(c) Approximate the total interest charges for a monthly payment of $ \$897.72 $ and for a monthly payment of $ \$1659.24 $.

(d) What is the vertical asymptote for the model?Interpret its meaning in the context of the problem.

Transcript

00:02 All right, here's an interesting problem to get you to think about mortgages and how much interest a person would pay over the life of a mortgage, how much money a person would save if they paid more each month, and so on.

00:14 So we have our function t, which represents the length of the mortgage in years as a function of the monthly payment.

00:22 And this particular equation is based on a $150 ,000 mortgage and a 6 % interest rate.

00:27 So let's find the length of the mortgage when the monthly payment is $897 .72.

00:34 So we're going to substitute that into our function.

00:41 Notice that there's x in the numerator and there's x in the denominator.

00:44 So we're going to put that number in both places.

00:47 And then we're going to put this into the calculator.

00:56 And we get t equals 30.

00:58 So that's in years.

00:59 So it would take 30 years to pay off the mortgage at that particular payment amount per month.

01:05 Now we do the same thing for a payment of $1 ,659 .24, almost twice the previous payment that we tried.

01:15 So that goes in for x.

01:18 Don't forget the natural log.

01:20 So that goes in for x in the quotient in both positions, and then into the calculator.

01:30 And let's see what we get.

01:32 And that one gives us 10.

01:34 So we get 10 years.

01:35 So by paying that extra money, the mortgage is paid in 10 years instead of 30 years...

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