Question

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the drawing shows two thermometers, a and b, whose temperatures are measured in °a and °b. the ice and boiling points of water are also indicated. (a) using the data in the drawing, determine the number of b degrees on the b scale that correspond to 1 °a on the a scale. (b) if the temperature of a substance reads +53.0 °a on the a scale, what would that temperature read on the b scale?
(a) 1 °a = number
(b) ( t_b ) = number
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Explanation:

Step1: Calculate the number of degrees between ice and boiling points for each scale

For scale A: \(60.0 - (- 30.0)=90.0\) \(^{\circ}A\)
For scale B: \(130.0 - 20.0 = 110.0\) \(^{\circ}B\)

Step2: Find the number of \(^{\circ}B\) per \(^{\circ}A\)

\(\frac{110.0}{90.0}=\frac{11}{9}\approx1.22\) \(^{\circ}B/^{\circ}A\)

Step3: Establish the linear relationship between \(T_A\) and \(T_B\)

Let \(T_B=mT_A + c\). Using the ice - point: when \(T_A=-30.0\), \(T_B = 20.0\). Substitute \(m=\frac{11}{9}\) into \(T_B=\frac{11}{9}T_A + c\)
\(20.0=\frac{11}{9}\times(-30.0)+c\)
\(c=20.0+\frac{11\times30.0}{9}=20.0+\frac{110}{3}=\frac{60 + 110}{3}=\frac{170}{3}\)

Step4: Calculate \(T_B\) when \(T_A = 53.0\)

\(T_B=\frac{11}{9}\times53.0+\frac{170}{3}\)
\(T_B=\frac{11\times53.0+170\times3}{9}=\frac{583+510}{9}=\frac{1093}{9}\approx121.4\) \(^{\circ}B\)

Answer:

(a) \(1.22\) \(^{\circ}B\)
(b) \(121.4\) \(^{\circ}B\)