Newton's Law of Cooling states that the rate of change of the temperature of an object, T, is proportional to the difference of T and the temperature of its surrounding environment. A pot of chili with temperature 23°C is placed into a −18°C freezer. After 2 hours, the temperature of the chili is 7°C.
Part A: Assuming the temperature T of the chili follows Newton's Law of Cooling, write a differential equation for T. (10 points)
Part B: What is the temperature of the chili after 4 hours? (20 points)
Part C: At what time, t, will the chili's temperature be −10°C? (10 points)
Answers
For this, let's go through each problem carefully and step-by-step.
According to the question, the rate of change of the temperature of any object that is defined by T, is directly proportional to the difference of T and the temperature of the environment around it, which we'll denote as X.
[tex]\frac{dT}{dt}[/tex][tex]= k (T-X)[/tex]
K is a constant of proportionality here. And the temperature of the surrounding environment is said to be (-18°C). Thus,
[tex]\frac{dT}{dt} = k(T+18)[/tex].
For part A, in order to find the differential equation for T, we need to solve for k. So we separate the variables and then integrate to solve the equation.
[tex]\int\limits{\frac{dT}{T+18} } = \int\limits {k} \, dt[/tex]
[tex]ln(T+18) = kt+c[/tex]
Now thw inital temperature of a pot of chili is 23°C, so at [tex]t = 0, T_0 = 23*C[/tex].
Substituting 23 for T and 0 for t, we have the following:
[tex]ln(23+18) = k(0)+c[/tex]
[tex]ln(41) = c[/tex]
We know the temperature of chili after 2 hours is 7°C, so we know that when [tex]t = 2, T_1 = 7[/tex]
Substituting t for 2, and T for 7, we get:
[tex]ln(7+18) = 2k+ln(41)[/tex]
[tex]ln(25) = 2k + ln(41)[/tex]
Solving for 2k
[tex]2k = ln(25) -ln(41)[/tex]
[tex]2k = ln(\frac{25}{41})[/tex]
[tex]k = \frac{1}{2}ln(\frac{25}{41})[/tex].
Substituting the value of [tex]\frac{dT}{dt} = k (T+18)[/tex], the differential equation obtained is [tex]\frac{dT}{dt} = \frac{1}{2}ln(\frac{25}{41})(T+18)[/tex].
For part B, to find the temperature of the chili after four hours, we first need to solve the above differential equation.
The solution of the differential equation is given by the equation [tex]ln(T+18) = kt+c[/tex]. Substituting the values of k and c, we have:
[tex]ln(T+18) = \frac{1}{2}ln(\frac{25}{41})t+ln(41)[/tex].
Using the above relation, at any time (t), the temperature (T) can be found out in the following.
At [tex]t = 4, T_2 = \phi[/tex]
[tex]ln(T_2+18)=\frac{1}{2}ln(\frac{25}{41})*4+ln(41)[/tex]
[tex]ln(T_2+18)=2ln(\frac{25}{41})+ln(41)[/tex]
[tex]ln(T_2+18)=-0.989 + 3.714[/tex]
[tex]ln(T_2+18)[/tex] ≅ [tex]2.725[/tex]
Solving the natural logarithm,
[tex]T_2+18 = e^{2.725} = 15.256[/tex]
[tex]T_2 =15.256 - 18[/tex]
[tex]T_2 = -2.744[/tex].
So the temperature of the chili after four hours would be -2.744°C approximately.
To find part C in what time the chili would be 10°C, we need to substitute again.
[tex]t = \phi[/tex][tex], T = -10[/tex]
[tex]ln(-10 + 18) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]
[tex]ln(8) = \frac{1}{2}ln(\frac{25}{41})t + ln(41)[/tex]
Solving for [tex]\frac{1}{2}ln(\frac{25}{41})t[/tex],
[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(8) - ln(41)[/tex]
[tex]\frac{1}{2}ln(\frac{25}{41})t = ln(\frac{8}{41})[/tex]
[tex]\frac{1}{2}ln(\frac{25}{41})t = -1.634[/tex]
[tex]ln(\frac{25}{41})t[/tex][tex]= -1.634 * 2[/tex]
[tex](-0.494)t=-3.268[/tex]
[tex]t = \frac{-3.268}{-0.494}[/tex]
[tex]t=6.615[/tex] hours, approximately.
Thus, the chili would reach -10°C at around 6.615 hours.
Hope this helped. This took me a long time.
Related Questions
WHEN LOIS MARTIN WAS BORN, HER FATHER DEPOSITED $2000 IN A SAVINGS
ACCOUNT IN HER NAME AT A SAVINGS AND LOAN ASSOCIATION (S&L). AT THE TIME,
THE S&L PAID 6% INTEREST COMPOUNDED SEMI-ANNUALLY. AFTER 10 YEARS, THE
S&L CHANGED TO A RATE OF 6% COMPOUNDED QUARTERLY. WHAT WAS THE
VALUE OF THE ACCOUNT AFTER 18 YEARS WHEN THE MONEY WAS WITHDRAWN TO
HELP PAY FOR HER COLLEGE EXPENSES
Answers
The future value of the savings account, after 18 years when it was withdrawn to help pay for Lois Martin's college expenses, would be $5,816.85.
How are the future values determined?The future values of the savings account are calculated in two installments.
The first installment is for 10 years when the account earns 6% compounded semiannually.
Using the future value after 10 years, the second installment is for 8 years when the account earns 6% compounded quarterly.
Future values can be determined using the future value formula or an online finance calculator, as follows:
Data and Calculations:Investment of $2,000 for 10 years:N (# of periods) = 20 (10 years x 2)
I/Y (Interest per year) = 6%
PV (Present Value) = $2,000
PMT (Periodic Payment) = $0
Results:
FV = $3,612.22
Total Interest $1,612.22
Investment of $3,612.22 for 8 years:N (# of periods) = 32 (8 years x 4)
I/Y (Interest per year) = 6%
PV (Present Value) = $3,612.22 ($2,000 + $1,612.22)
PMT (Periodic Payment) = $0
Results:
FV = $5,816.85
Total Interest $2,204.63
Thus, the value of the account after 18 years was $5,816.85.
Learn more about future values at https://brainly.com/question/989421
#SPJ1
how is the e - lambda figured out as 0.00248?
Answers
Answer:
Step-by-step explanation:
e^-6 = 1/e^6 = about 0.00248
What is the value of 2 + (- 2/3)^2 ÷ 1/3 ?
16
3
-2
0
Answers
Answer:
3
Step-by-step explanation:
2+ (-2/3)^2 ÷ 1/3
make -2/3 postive because its being raised by an even exponent
you never divide by fractors so make 1/3 into 3 and make it multiplication
2+ (2/3)^2 × 3
raise the fraction to the power of 2
2+ (4/9)× 3
cancel out the GCF
2+ 4/3
add
10/3
simplify
3.3 ≅ 3
A chemical company makes two brands of antifreeze. The first brand is 30% pure antifreeze and the second brand is 55% pure antifreeze. In order to obtain 160 gallons of a mixture that contains 45% pure antifreeze, how many gallons of each brand of antifreeze must be used?
Answers
Answer:
64 gallons 30%96 gallons 55%Step-by-step explanation:
The amount of each brand can be found by writing and solving an equation that makes use of the given relations.
SetupLet x represent the quantity of 55% pure antifreeze needed in the mixture. Then the quantity of the 30% brand is (160-x). The amount of pure antifreeze in the mixture is ...
0.55x +0.30(160 -x) = 0.45(160)
SolutionSubtracting 0.30(160), we have ...
x(0.55 -0.30) = 160(0.45 -0.30)
x = 24/0.25 = 96 . . . . . simplify, divide by the coefficient of x
96 gallons of the 55% brand, and 64 gallons of the 30% brand must be used.
__
Read more about mixture problems here:
https://brainly.com/question/26751698
Show all work to write the expression in simplified rational exponent form:
[tex]\sqrt[4]{17x^{2}}[/tex]
(Fourth root of 17 x squared)
Answers
Answer:
[tex](17x)^{\frac{2}{4}}[/tex]
Step-by-step explanation:
There's not much work to show for this problem, but I'll explain it for you
Use the exponent rule for roots to combine the 4 and the 2 like a fraction
2 on the top and 4 on the bottom
[tex](17x)^{\frac{2}{4} }[/tex]
That's it! Pretty easy
Good Luck!
pls mark brainliest
Reagan has a part-time job walking dogs. She makes $12.20 per hour. If she earns $106.75 this week, how many hours does she work?
Answers
Step by step. $106.75 earned divided by $12.20 per hour = 8.75 hours worked.
Answer:
8.75
Step-by-step explanation:
106.75 divided by 12.20= 8.75
Question 8 of 10
Factor the polynomial: -5x³ - 10x² - 15x
OA. 5x(x²+2x-3)
OB. -5x(-x2 - 2x - 3)
C. -5x(x2 + 2x - 15)
OD. -5x(x2+2x+3)
Answers
All terms are negative, so A won’t work
-5x^3/-5x = x^2, so B won’t work
-15x/-5x = 3, so C won’t work
100 000 , 10 000 , 1000 , 100 , A , B
The two missing numbers in the pattern are:
A=___?
B=___?
Answers
The two missing numbers in the pattern are 10 and 1
How to determine the missing numbers?The sequence is given as:
100 000 , 10 000 , 1000 , 100 , A , B
From the above sequence, the next term is the quotient of the current term and 10.
So, we have:
A = 100/10 = 10
B = A/10 = 10/10 = 1
Hence, the two missing numbers in the pattern are 10 and 1
Read more about sequence at:
https://brainly.com/question/21961097
#SPJ1
Question 1 Which number has the greatest value? O 0.095 O 1.24 1.215 O 0.365 Confident
Answers
Answer:
1.24
Step-by-step explanation:
because 1.24>1.215>0.365>0.095
There are 96 football players at all levels at Davenport High School. If two-thirds of the players make the honor roll for their grades how many olayers are on the list?
64 players
48 players
72 players
54 players
Answers
Reason- 2/3 of 96 = (32 x 2 = 64) telling us how many got on the list
Sports clubs have increasingly been acknowledged as important settings for health promotion. The term sport club is synonymous with organized sport and both are considered to consist of environments that support the healthy behaviors of its athletic members. One of the member Sanjaya pays rupees 500 in advance on his account at a sports club ,each time he visits the club 10 rupees is deducted from his account. Q6. Which of the following equation represent the left balance of sanjaya’s y after visiting x number of days? * A) y=500+10x B) y=10+500x C) x=500-10y D) y=500-10x NO RESPONSE
Answers
The linear equation that represent the left balance of sanjaya’s y after visiting x number of days is:
y = 10x + 500.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.From the data given, we have that:
The y-intercept is of 500.The slope is of 10.Hence the balance is given as follows:
y = 10x + 500.
More can be learned about linear functions at https://brainly.com/question/24808124
#SPJ1
List the angles in order from the largest to the smallest for ΔABC. AB⎯⎯⎯⎯⎯⎯⎯=14, AC⎯⎯⎯⎯⎯⎯⎯⎯=15, BC⎯⎯⎯⎯⎯⎯⎯⎯=16
Answers
Answer:
BC 1
AC 2
AABC 3
Step-by-step explanation:
BC=16
AC = 15
AABC= 15
Answer: C, B, A
Step-by-step explanation: It just is
If 2 angles have the same angle measure they are said to be what
Answers
2 angles that have the same angle measure are congruent.
Given that the two angles have the same angle measure.
Congruence is a term reserved for geometry. Two metrics are congruent when you can perfectly map to each other by mirroring, rotating, and translating without distortion.
We know that two shapes or objects are congruent if they have the same shape and the same size. For two angles to be congruent, they must coincide when they are overlapped. This is only possible if the two angles are equal.
Therefore, if two angles have the same measure, then they are said to be congruent.
Learn more about congruent angle from here brainly.com/question/2938476
#SPJ1
Ronaldo specializes in creating sewing patterns for dresses, and wants to set up his own website to sell electronic copies. He pays $120 for a domain name and web hosting services for the first year. Ronaldo makes $14.25 for each electronic pattern that he sells. How many patterns must he sell to cover the cost of the website for the first year? Round your answer to a whole number.
Answers
Answer:
8
Step-by-step explanation:
120/14.25=8.4 rounded off to a whole number is 8
Graph on the number line all values of x that satisfy 2x+2/x+7 >= 0
Please help, if you help me I will give you a brainlist.
Answers
All values of x that satisfy 2x+2/x+7 >= 0 are represented as x >= -1
How to graph the inequality on a number line?The inequality expression is given as:
2x+2/x+7 >= 0
Multiply both sides by x + y
2x+2 >= 0
Subtract 2 from both sides
2x > -2
Divide both sides by 2
x >= -1
Hence, all values of x that satisfy 2x+2/x+7 >= 0 are represented as x >= -1
See attachment for the number line of x >= -1
Read more about number lines at:
https://brainly.com/question/4727909
#SPJ1
Juan and Ari went to the farmers market to buy fruit. Juan’s bag of apples that he bought weighed StartFraction 31 over 7 EndFraction pounds. Ari’s bag of oranges that he bought weighed StartFraction 19 over 3 EndFraction pounds. Using compatible fractions, estimate how much heavier Ari’s bag is compared to Juan’s bag.
Answers
Answer:
approximately 2 lb
Step-by-step explanation:
Juan: 31/7
Ari: 19/3
Juan's fraction is close to 32/8 = 4
Ari's fraction is close to 18/3 = 6
6 - 4 = 2
Answer: approximately 2 lb
Find the point at which the line with parametric equations x = 2 3t, y =-4t, z =5 t intersects the plane 4x 5y -2z=18
Answers
It looks like you have
[tex]\begin{cases}x=2+3t\\y=-4t\\z=5+t\end{cases}[/tex]
Substituting these in to the plane equation, we have
[tex]4x+5y-2z = 4(2+3t) + 5(-4t) - 2(5+t) = -2-10t = 18 \\\\ \implies-10t = 20 \implies t = -2[/tex]
When [tex]t=-2[/tex], we get the point
[tex]\begin{cases}x=2+3(-2)\\y=-4(-2)\\z=5+(-2)\end{cases} \implies (x,y,z) = \boxed{(-4,8,3)}[/tex]
Two sides of a triangle are 12cm and 8cm. Complete the inequality to show the possible lengths for the third side.
Answers
The inequalities that represents the third side of the triangle is c ≥ 4 or c ≤ 20
How to find the third side of a triangle?The triangle inequality theorem states that that the sum of any two sides of a triangle is greater than or equal to the third side.
Therefore, the two sides of the triangle are 12 cm and 8 cm.
A triangle with sides a, b and c follows the principle below;
a + b ≥ c.
Therefore,
let
c = third sides
12 + 8 ≥ c
20 ≥ c
c + 8 ≥ 12
c ≥ 4
c + 12 ≥ 8
Therefore, the third third should be greater than or equals to 4 or less than or equals to 20
c ≥ 4 or c ≤ 20
learn more on triangle here: https://brainly.com/question/28105571
#SPJ1
Question 5
Find the interest rate for a principal of $6570 and charged $30105 in interest for 12 years.
The Formula is
I
PT
x 100
Take the interest divide by the principal
then divide by the years
then times by 100
Round your answer to 1 decimal place.
Answers
Answer:
R=i
Step-by-step explanation:
R=i×100÷p×t
r=30105×100÷6570×12
r=3010500÷78840
r=38.18
Find the number if 11% of it is 55
Answers
Answer:
Step-by-step explanation:
.11x = 55
11x = 5500
x = 500
Which of the following would be an acceptable first step in simplifying the expression
sinx/1- cosx?
A. sinx - sinx/1- cos x
B. 1/cscx - cosx
C. Cannot be simplified further
D. sin x(1+cosx)/(1-cosx)(1+cosx)
Answers
Answer: A
Step-by-step explanation: Because it can be simplified twice.
Five interlocked, colored rings make up the Olympic symbol. The rings are
blue, black, red, yellow, and green and represent Africa, the Americas, Asia,
Australia, and Europe. The symbol looks like this:
1. In how many ways could the five colors of the Olympic symbol have been
arranged among the rings?
Answers
Answer:
120
Step-by-step explanation:
5 rings 5 colors
for the first ring there are 5 colors to choose from
4 colors for the second
3 for the third
2 for the fourth
1 for the last one
so 5! = 120
Merina is scheduled to make two loan payments to Bradford in the amount of $1,000 each, two months and nine months from now. Merina doesn't think she can make those payments and offers Bradford an alternative plan where she will pay two equal amounts seven months from now followed by another equal payment seven months later. Bradford determines that 8.5% is a fair interest rate. Using the second unknown equal payment as your focal date, what are the equal payments?
Answers
well, we're assuming all along that Merina owes Bradford $2000, because in the 1st scenario, she was going to pay twice $1000.
on the 2nd scenario, she'll be paying the same $2000 but split 7 months from now and then 7 months later, same 2000 bucks, at which point Bradford applied 8.5% interest.
using those assumptions, since the wording is not quite clear, we can say that Merina is simply paying 2000 bucks plus the 8.5%
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{8.5\% of 2000}}{\left( \cfrac{8.5}{100} \right)2000}\implies 170 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\stackrel{principal}{2000}~~ + ~~\stackrel{interest}{170}}{2}\implies \stackrel{\textit{two equal payments of}}{1085}[/tex]
Find the cube root of Z =1+i the power of 5
Answers
Answer:
Cube Root Calculator
Use this calculator to find the cube root of positive or negative numbers. Given a number x, the cube root of x is a number a such that a3 = x. If x positive a will be positive, if x is negative a will be negative. Cube roots is a specialized form of our common radicals calculator.
It will help you alot.
Step-by-step explanation:
question:
|x-2|, if x > 5
simplify without the absolute value sign
Answers
Answer:
x - 2, if x > 5
Step-by-step explanation:
The vertical lines either side of the expression mean absolute value.
The absolute value of a number is its positive numerical value.
if x > 5 then as 5 > 2, the values inside the vertical lines will always be positive. Therefore, we can disregard the absolute value.
Therefore:
x - 2, if x > 5
To find the range (output values) of the expression, substitute x = 5 into the expression:
⇒ 5 - 2 = 3
Therefore, |x - 2| > 3, if x > 5
Compute the probability of event E if the odds in favor of E are 28 to 29.
Answers
Answer:
28/57
Step-by-step explanation:
odds in favor of event occurring = (probability event occurs)/(probability event does not occur)
Let p = probability event occurs.
Then the probability the event does not occur is 1 - p.
Let the odds = a/b.
a/b = p/(1 - p)
a(1 - p) = bp
a - ap = bp
bp + ap = a
p(a + b) = a
p = a/(a + b)
p = 28/(28 + 29)
p = 28/57
As per the given scenario, the probability of event E is 28/57 or approximately 0.4912.
What is probability?Probability is a measure of the likelihood or chance of an event occurring. It is usually expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to occur.
If the odds in favor of an event E are "a to b," then the probability of E is given by:
P(E) = a / (a + b)
In this case, the odds in favor of E are 28 to 29, which means that there are 28 favorable outcomes for E and 29 unfavorable outcomes against E.
So, using the formula above, we can compute the probability of E as:
P(E) = 28 / (28 + 29)
P(E) = 28 / 57
Therefore, the probability of event E is 28/57 or approximately 0.4912.
For more details regarding probability, visit:
https://brainly.com/question/30034780
#SPJ2
Hi,
Could anyone help with this please?
Many thanks,
Answers
Length: 2.6 meters
Width: 1.8 meters
Area: 4.68 meters
Since the scale is 1/40, we can multiply the centimeter measurements by 40 thus making it our scale factor.
Length
2.6 x 40 = 260
260 cm = 2.6 m
Width
4.5 x 40 = 180
180 cm= 1.8 m
Area
1.8 x 2.6 = 4.68 m
Length: 2.6 meters
Width: 1.8 meters
Area: 4.68 meters
According to a recent poll at a university,70.7 % of high school seniors in a certain region had a driver's license. A sociologist thinks this rate has declined. The sociologist surveys randomly selected high school seniors and finds that have a driver's license. a. Pick the correct null hypothesis. b. Pick the correct alternative hypothesis. c. In this context, what does the symbol p represent?
Answers
your mother i swear
Step-by-step explanation:
ask yo mom
on a unit circle, the terminal point of 0 is (1/2, square root 3/2). what is sin 0?
A. square root 3/2
B. 1/2
C. square root 3/3
D. square root 3
Answers
Answer: A
Step-by-step explanation:
[tex]\sin \theta[/tex] is equal to the y-coordinate.
Answer: square root 3/2
Step-by-step explanation:
Please I need the answer this is due in on monday!!
Answers
The number sets that each number belong too are given as follows:
1. Rational.
2. Irrational.
3. Rational.
4. Whole.
5. Whole.
6. Integer.
What is the classification of the number -4.2?It is a terminating decimal, hence it is a rational number.
What is the classification of the number 3 times square root of 5?The square root of 5 is not exact, hence it is a non-terminating decimal, meaning that is an irrational number.
What is the classification of the number 5/3?It is a fraction, hence it is a rational number.
What is the classification of the number 9?It is a positive non-decimal number, hence it is a whole number.
What is the classification of the square root of 16?The square root of 16 is of 4, which is a positive non-decimal number, hence it is a whole number.
What is the classification of the square root of -8/2?The result of the division is of -4, which is a negative non-decimal number, hence it is an integer number.
More can be learned about the number sets at https://brainly.com/question/10814303
#SPJ1
