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NeTakaya
2 years ago
7

Solve by using proper methods. Show your work.

Mathematics
1 answer:
NeX [460]2 years ago
7 0

Answer:

$70,201.38

Step-by-step explanation:

To determine the principal amount, we can use the formula:

P=\dfrac{A}{(1+\frac{r}{n})^{nt}}

A = $200,000

r = 7% or 0.07

t = 15 years

n = 12

Now let's substitute our values.

P=\dfrac{200,000}{(1+\frac{0.07}{12})^{12(15)}}

P=\dfrac{200,000}{(1.00583333333)^{12(15)}}

P=70,201.38

So the principal needed to get 200,000 in 15 years is $70,201.38.

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Answer:

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Step-by-step explanation:

The core temperature of the object after 4 hours can be found using an exponential decay formula to model the decay of the difference between core temperature and ambient.

<h3>Cooling Model</h3>

The solution to the differential equation described by Newton's law of cooling is the exponential equation ...

  y = ab^t +c

where 'a' is the initial core temperature difference from ambient, 'b' is the decay factor of that difference in 1 unit of time period t. 'c' is the ambient temperature.

For this problem, the ambient temperature is c=80, and the differences of interest are ...

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Using these values in the model gives ...

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Note that units of time are hours.

<h3>Application</h3>

We want y when t=4.

  y = 1120(75/112)^4 +80 ≈ 1120(0.20108) +80 ≈ 305.212

The core temperature after 4 hours is about 305 °F.

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<em>Additional comment</em>

The differential equation will have a solution of the form ...

  T-T_R=(T_0-T_R)e^{kt}

where k = ln(75/112) ≈ -0.40101

In the above, we defined b = e^k = 75/112. Accuracy with this fraction can be better than using a truncated value of k.

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