Mathematics, 04.07.2021 02:10 xxxamslashxxx9

The temperature of a cup of coffee varies according to Newton's Law of Cooling: -"dT/dt=k(T-A), where is the temperature of the coffee, A is the room temperature, and k is a positive constant. If the coffee cools from 100°C to 90°C in 1 minute at a room temperature of 25*C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes,
74
67
60
42

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Answer from: gaby4567

B) 67°C.

Step-by-step explanation:

Newton's Law of Cooling is given by:

$\displaystyle \frac{dT}{dt}=k(T-A)$

Where T is the temperature of the coffee, A is the room temperature, and k is a positive constant.

We are given that the coffee cools from 100°C to 90°C in one minute at a room temperature A of 25°C.

And we want to find the temperature of the coffee after four minutes.

First, solve the differential equation. Multiply both sides by dt and divide both sides by (T - A). Hence:

$\displaystyle \frac{dT}{T-A}=k\, dt$

Take the integral of both sides:

$\displaystyle \int \frac{dT}{T-A}=\int k\, dt$

Integrate:

$\displaystyle \ln\left|T-A\right| = kt+C$

Raise both sides to e:

$|T-A|=e^{kt+C}=Ce^{kt}$

The temperature of the coffee T will always be greater than or equal to the room temperature A. Thus, we can remove the absolute value:

$\displaystyle T=Ce^{kt}+A$

We are given that A = 25. Hence:

$\displaystyle T=Ce^{kt}+25$

Since the coffee cools from 100°C to 90°C, the initial temperature of the coffee was 100°C. Thus, when t = 0,T = 100:

$100=Ce^{k(0)}+25\Rightarrow C=75$

Hence:

$T=75e^{kt}+25$

We are given that the coffee cools from 100°C to 90°C after one minute at a room temperature of 25°C.

So, T = 90 given that t = 1. Substitute:

$90=75e^{k(1)}+25$

Solve for k:

$\displaystyle e^k=\frac{13}{15}\Rightarrow k=\ln\left(\frac{13}{15}\right)$

Therefore:

$\displaystyle T=75e^{\ln({}^{13}\! /\!{}_{15})t}+25$

Then after four minutes, the temperature of the coffee will be:

$\displaystyle \begin{aligned} \displaystyle T&=75e^{\ln({}^{13}\! /\!{}_{15})(4)}+25\\\\&\approx 67^\circ\text{C}\end{aligned}$

Hence, our answer is B.

Answer from: Quest

I’m sorry i can’t u

Answer from: Quest

$a_1 = 1$ , and

$a_{n+1} = a_n + (n + 1)$ for $n \ge 1$

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote $a_1 = 1$ , $a_2 = 3$ , $a_3 = 6$ , so that we can call

1 the 1st term, 3 the 2nd term, 6 the 3rd term,etc.

note the following:

from the 1st term (1) to the 2nd term (3), we go up by 2,from the 2nd term (3) to the 3rd term (6), we go up by 3from the 3rd term (6) to the 4th term (10), we go up by 4, andfrom the 4th term (10) to the 5th term (15), we go up by 5.

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from $a_1 = 1$ , to get to $a_2 = 3$ , we add $2(=1+1)$

from $a_2 = 3$ , to get to $a_3 = 6$ , we add $3 (=2+1)$

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

$a_1 = 1$ , and $a_{n+1} = a_n +(n + 1)$ for $n \ge 1$

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The temperature of a cup of coffee varies according to Newton's Law of Cooling: -"dT/dt=k(T-A), wher...

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