A cold yam is put into a hot oven, the temperature of the yam begins to rise. The rate R in degrees per minute at which the temperature rises is governed by Newton's Law of Heating, which says that the rate is proportional to the temperature different between the yam and the oven. If the oven is at 350 degrees and the temperature of the yarn H degrees F.

Question

A cold yam is put into a hot oven, the temperature of the yam begins to rise. The rate R in degrees per minute at which the temperature rises is governed by Newton's Law of Heating, which says that the rate is proportional to the temperature different between the yam and the oven. If the oven is at 350 degrees and the temperature of the yarn H degrees F.

a) write a formula giving R as a function of H

i know that the proportion formula is y=kx. would i put the 350 in for y and leave x as the variable? i don't know what to do. thank you.

Answer 1

The question tells you that the rate (R) is proportional to the temp difference (D) between the yam and the surroundings.

so R is proportional to D

or R=kD

if the temperature of the yam is H and the temperature of the surroundings is 350 then

D=350-H

so R=k(350-H)

which makes sense as the rate of heating will approach zero as H approaches 350.

Answer 2

To write a formula for R as a function of H, you need to express the rate of temperature rise (R) as a proportion of the temperature difference between the yam and the oven (H - 350). Since Newton's Law of Heating states that the rate is proportional to this temperature difference, you can use the proportion formula y = kx.

In this case, y represents R (the rate of temperature rise) and x represents (H - 350) (the temperature difference). The constant of proportionality is k, which we need to determine.

To find k, you can use the given information that the rate of temperature rise is governed by Newton's Law of Heating. In this case, it implies that when H = 350, R = 0 (since there is no temperature difference, the temperature will not rise). We can substitute these values into the proportion formula:

0 = k(350 - 350)

Simplifying,

0 = k(0)

This equation tells us that when the temperature difference is zero, the rate of temperature rise is also zero. Therefore, k must be 0 as well.

Now, substitute k = 0 back into the proportion formula:

R = 0(H - 350)

Which simplifies to:

R = 0

So, the formula for R as a function of H is simply R = 0. This means that the rate of temperature rise is constant and always equal to zero, regardless of the temperature of the yam (H).

Answer 3

To write the formula giving R as a function of H, we can utilize Newton's Law of Heating, which states that the rate of temperature change is proportional to the temperature difference between the yam and the oven.

Let's denote the proportionality constant as k.

So, the formula for R as a function of H can be written as:

R = k(H - 350)

Here, R represents the rate at which the yam's temperature is rising (in degrees per minute), and H represents the temperature of the yam (in degrees Fahrenheit). The temperature of the oven, which is 350 degrees Fahrenheit, is subtracted from H to determine the temperature difference between the yam and the oven.

Please note that k is the constant of proportionality, which could be determined experimentally or contextually based on the specific heat transfer characteristics of yams or the material being heated.