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.)

a.  k = -0.01014 s⁻¹

b.  

c.  

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

Taking the integral.

㏑ (T -60) = kt + C

T - 60 =

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

T(20) = 120

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

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

T(60) = 100

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

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

- 40 k = 0.4054651

k = -0.01014 s⁻¹

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

Recall that :

Since y is the temperature of the body , then :

(c) What is the formula for y(t)?

From equation (1) ;

where;

Let y be measured in degrees Fahrenheit

From equation (2)

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

At the moment it is submerged t = 0

y(t) = 60 + 70.485

y(t) = 130.485°F

irrational, cannot