Mathematics, 05.08.2020 04:01 huntbuck14

A hot metal bar is submerged in a large reservoir of water whose temperature is 60°F. The temperature of the bar 20 s after submersion is 120°F. After 1 min submerged, the temperature has cooled to 100°F. (Let y be measured in degrees Fahrenheit, and t be measured in seconds.) (a) Determine the cooling constant k. k = s−1 (b) What is the differential equation satisfied by the temperature y(t)? (Use y for y(t).) y'(t) = (c) What is the formula for y(t)? y(t) = (d) Determine the temperature of the bar at the moment it is submerged. (Round your answer to one decimal place.)

Answers: 1

Show answers

Answers

Answer from: Ashley606hernandez

a. k = -0.01014 s⁻¹

b. $\mathbf{\dfrac{dy}{dt} = - \dfrac{In(\dfrac{3}{2})}{40}(y-60)}$

c. $\mathbf{y(t) = 60+ \dfrac{60 \sqrt{3}}{\sqrt{2}} \ e^{\dfrac{-In(\dfrac{3}{2})\ t}{40}}}$

d. y(t) = 130.485°F

Step-by-step explanation:

A hot metal bar is submerged in a large reservoir of water whose temperature is 60°F. The temperature of the bar 20 s after submersion is 120°F. After 1 min submerged, the temperature has cooled to 100°F.

(Let y be measured in degrees Fahrenheit, and t be measured in seconds.)

We are to determine :

a. Determine the cooling constant k. k = s−1

By applying the new law of cooling

$\dfrac{dT}{dt} = k \Delta T$

$\dfrac{dT}{dt} = k(T_1-T_2)$

$\dfrac{dT}{dt} = k (T - 60)$

Taking the integral.

$\int \dfrac{dT}{T-60} = \int kdt$

㏑ (T -60) = kt + C

T - 60 = $e^{kt+C}$

$T = 60+ C_1 e^{kt} ---- (1)$

After 20 seconds, the temperature of the bar submersion is 120°F

T(20) = 120

From equation (1) ,replace t = 20s and T = 120

$120 = 60 + C_1 e^{20 \ k}$

$120 - 60 = C_1 e^{20 \ k}$

$60 = C_1 e^{20 \ k} --- (2)$

After 1 min i.e 60 sec , the temperature = 100

T(60) = 100

From equation (1) ; replace t = 60 s and T = 100

$100 = 60 + c_1 e^{60 \ t}$

$100 - 60 =c_1 e^{60 \ t}$

$40 =c_1 e^{60 \ t} --- (3)$

Dividing equation (2) by (3) , we have:

$\dfrac{60}{40} = \dfrac{C_1e^{20 \ k } }{C_1 e^{60 \ k}}$

$\dfrac{3}{2} = e^{-40 \ k}$

$-40 \ k = In (\dfrac{3}{2})$

- 40 k = 0.4054651

$k = - \dfrac{0.4054651}{ 40}$

k = -0.01014 s⁻¹

b. What is the differential equation satisfied by the temperature y(t)?

Recall that :

$\dfrac{dT}{dt} = k \Delta T$

$\dfrac{dT}{dt} = \dfrac{- In (\dfrac{3}{2})}{40}(T-60)$

Since y is the temperature of the body , then :

$\mathbf{\dfrac{dy}{dt} = - \dfrac{In(\dfrac{3}{2})}{40}(y-60)}$

From equation (1) ;

where;

$T = 60+ C_1 e^{kt} ---- (1)$

Let y be measured in degrees Fahrenheit

$y(t) = 60 + C_1 e^{-\dfrac{In (\dfrac{3}{2})}{40}t}$

From equation (2)

$C_1 = \dfrac{60}{e^{20 \times \dfrac{-In(\dfrac{3}{2})}{40}}}$

$C_1 = \dfrac{60}{e^{-\dfrac{1}{2} {In(\dfrac{3}{2})}}}$

$C_1 = \dfrac{60}{e^ {In(\dfrac{3}{2})^{-1/2}}}}$

$C_1 = \dfrac{60}{\sqrt{\dfrac{2}{3}}}$

$C_1 = \dfrac{60 \times \sqrt{3}}{\sqrt{2}}}$

$\mathbf{y(t) = 60+ \dfrac{60 \sqrt{3}}{\sqrt{2}} \ e^{\dfrac{-In(\dfrac{3}{2})\ t}{40}}}$

(d) Determine the temperature of the bar at the moment it is submerged.

At the moment it is submerged t = 0

$\mathbf{y(0) = 60+ \dfrac{60 \sqrt{3}}{\sqrt{2}} \ e^{\dfrac{-In(\dfrac{3}{2})\ 0}{40}}}$

$\mathbf{y(t) = 60+ \dfrac{60 \sqrt{3}}{\sqrt{2}} }$

y(t) = 60 + 70.485

y(t) = 130.485°F

Answer from: Quest

i'm not sure what you mean but in todays world there is a lot of women that are the bread winners in the family. there is a lot more single mothers now that are going back to get an education which is boosting the economy. i believe there has been a boost in the economy due to women the cosmetic industry is boosting, chemist, the sky is becoming the limit for women in todays economy. most women are also putting off having kids just to make sure they have a education. i don't know how these two relate though to scriptures because it's not the way the lord wanted it. the man is suppose to be the bread winner and due to social media and the easy access we have there is a lot of cheating going on.

step-by-step explanation:

Answer from: Quest

i dont know it

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 17:30

When a rectangle is dilated, how do the perimeter and area of the rectangle change?

Answers: 2

Answer

Mathematics, 21.06.2019 20:10

Suppose g(x) = f(x + 3) + 4. which statement best compares the graph of g(x) with the graph of f(x)?

Answers: 2

Answer

Mathematics, 21.06.2019 21:30

Madeline takes her family on a boat ride. going through york canal, she drives 6 miles in 10 minutes. later on as she crosses stover lake, she drives 30 minutes at the same average speed. which statement about the distances is true?

Answers: 3

Answer

Mathematics, 21.06.2019 22:30

The dash figure is a dilation if the original figure. find the scale factor. classify the dilation as an enlargement or a reduction

Answers: 2

Answer

You know the right answer?

A hot metal bar is submerged in a large reservoir of water whose temperature is 60°F. The temperatur...

Questions

Mathematics, 23.06.2021 23:20

PLEASE HELP URGENTGG ...

Mathematics, 23.06.2021 23:20