A botanist secures a 30-year mortgage for $513,000 at an annual interest rate of 4.175% with 1.4 points. The loan origination fe
1 answer:
Answer:
4.346%
Step-by-step explanation:
The points and origination fee are assumed to be capitalized, so the loan payment amount is based on the total of the original loan amount and those fees. The APR is calculated as the rate that would be charged on the original loan value to produce that payment.
<h3>Loaned amount</h3>The loaned amount is the sum of the original loan value and the added points.
P = $513,000×(1 +1.4% +0.65%) = $523,516.50
<h3>Payment</h3>The monthly payment on this amount is given by the amortization formula ...
A = P(r/12)/(1 -(1 +r/12)^(-12t))
where r=0.04175, the annual interest rate, t=30, the period in years
A = $523,516.50(0.04175/12)/(1 -(1 +0.04175/12)^-360)) ≈ $2552.45
<h3>APR</h3>There is no formula for calculating the APR. It must be developed iteratively or graphically. The attached calculator images show a couple of different ways the value can be found.
The payment of $2552.45 on a loan value of $513,000 represents an APR of 4.346%.
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Step-by-step explanation:
1. secx / cscx
2. sinx /cosx
Observe attached picture.
On picture we have:
A = height of flagpole = x ft
B = length of flagpole's shadow = 24 ft
C = height of sign = 6 ft
D = length of sign's shadow = 3 ft
When we draw a picture representing this problem we can also add another line marked in red. This way we can see that we have two right-angle triangles. We can see that both have same angle marked with α.
We can apply trigonometry rules to find height of flagpole.
From small triangle containing sign we can find tangens function:
Similarly we can do for large triangle containing flagpole:
We see that these two equations have same left sides. This means that their right sides must also be same:
We can solve for A:
Height of flagpole is 48 feet.
On picture we have:
A = height of flagpole = x ft
B = length of flagpole's shadow = 24 ft
C = height of sign = 6 ft
D = length of sign's shadow = 3 ft
When we draw a picture representing this problem we can also add another line marked in red. This way we can see that we have two right-angle triangles. We can see that both have same angle marked with α.
We can apply trigonometry rules to find height of flagpole.
From small triangle containing sign we can find tangens function:
Similarly we can do for large triangle containing flagpole:
We see that these two equations have same left sides. This means that their right sides must also be same:
We can solve for A:
Height of flagpole is 48 feet.