Question

A platinum resistance thermometer has a resistance of 10.40 ohms at00 C and 14.35 ohms at 100 C. Assuming that the resistance changesuniformly with temperature, what is(a) The temperature when the resistance is 11.19 ohms?(b) The resistance of the thermometer when the temperature is 45 C?

Answer 1

(a) 0.05 ° C

(b) 12.17 ohms

Step-by-step explanation

Assuming that the resistance changes uniformly with temperature we can set the equation of the line

so

Step 1

find the equation of the line:

the equation of a line has the form:

`\begin{gathered} y=mx+b \\ where\text{ m is the slope} \\ b\text{ is the y-intercept} \end{gathered}`

to do that, we need 2 points from the line, so let

x represents the temperature

y represents resistance

`\begin{gathered} P1(0,10.40) \\ P2(100,14.35) \end{gathered}`

now, to find the slope of a line we need to use the expression

`\begin{gathered} slope=(y_2-y_1)/(x_2-x_1) \\ where \\ P1(x_1,y_1) \\ P2(x_2,y_2) \end{gathered}`

then, replace the values in the formula

`slope=(14.35-10.4)/(100-0)=(3.95)/(100)=0.0395`

so, the slope is 0.0395

b) now, use the slope point formula to find the equation of the line

`\begin{gathered} y-y_1=m(x-x_1) \\ where\text{ m is the slope and P1\lparen x}_1,y_1) \\ replace \\ y-10.4=0.0395(x-0) \\ y=0.0395x+10.4 \end{gathered}`

so, the equation of the lines is

y=0.0395x+10.4

Step 2

Now, we can use the equation to find the asked values

(a) The temperature when the resistance is 11.19 ohms?

let

`\begin{gathered} resistance\text{ = 11.19,so y=11.19} \\ replace\text{ in the equation and solve for x} \\ y=0.0395x+10.4 \\ 11.19=0.0395x+10.4 \\ 11.19-10.4=0.0395x \\ 0.79=0.0395x \\ x=(0.0395)/(0.79)=0.05 \end{gathered}`

therefore

(a) 0.05 ° C

Step 3

(b) The resistance of the thermometer when the temperature is 45 C?

let

`tempereature\text{ = x= 45 C}`

now, replace in the expression to find the resistance

`\begin{gathered} y=0.0395x+10.4 \\ y=0.0395(45)+10.4 \\ y=12.17 \end{gathered}`

therefore,

(b) 12.17 ohms

I hope this helps you