Question

A piece of metal is heated to 428°C before it is left to cool. The temperature of the metal falls at a rate of 23°C per minute for the first 3 minutes, at a rate of 15°C per minute for the next 15 minutes and then at a rate of 8°C per minute until it reaches room temperature of 25°C. What will be the temperature of the metal after 9 minutes, and what will be the time needed for the metal to reach a temperature of 25°C ?

Answer 1

Answer:

1) Temperature of the metal after 9 minutes =  269°C

2) $Required\ temperature = 31\ min\ 37.5\ sec$

Step-by-step explanation:

Given:

Final temperature of the metal = 428°C

Temperature falls per minute for 3 minute  = 23°C

Temperature falls per minute for next 15 minute  = 15°C

Temperature falls per minute until room temperature  = 8°C

We need to find the temperature after 9 minutes and what will be the time needed for the metal to reach a temperature of 25°C.

Solution:

First part:

As per given statement, the temperature of the metal falls at a rate of 23°C per minute for the first 3 minutes, so temperature falls in 3 minutes.

Temperature falls per minute  = 23°C

Therefore,

$Temperature\ falls\ for\ 3\ minute = 23\times 3$

Temperature falls for 3 minute = 69°C

After that temperature falls 15°C per minute for next 15 minute, so it is falls in  

6 minutes.

Temperature falls per minute after 3 minutes  = 15°C

Therefore,

$Temperature\ falls\ for\ 6\ minutes\ after\ 3\ minutes = 15\times 6$

Temperature falls for 6 minutes after 3 minutes = 90°C

So temperature falls in 9 minutes = 90 + 69 = 159°C

So the temperature of the metal after 9 minutes = 428 - 159 = 269°C

Second part:

Temperature falls 23°C per minute for 3 minutes = 69°C

Temperature falls 15°C per minute for 15 minute = 225°C

So, leaving temperature after 18 minutes = Top temp - temp falls in 18 minutes.

Leaving temperature after 18 minutes = 428 - (69 + 225)

Leaving temperature after 18 minutes = 428 - (69 + 225)

Leaving temperature after 18 minutes = 134°C

Temperature falls 8°C per minute until it reaches room temperature of 25°C = 134 - 25 = 109°C

So the rate of the falling temperature = $\frac{109}{8}= 13\frac{5}{8}=13\ min\ 37.5\ sec$

$Required\ temperature = 31\ min\ 37.5\ sec$

Therefore, the required temperature of the metal to reach 25°C is equal to 31 min 37.5 sec.