Answer
a. $^{\circ}$C = $\frac{14}{5}$($^{\circ}$X) - 10.
b. 11.4 $^{\circ}$X
c. 152.4 $^{\circ}$C, 425.6 $^{\circ}$K, 306.3 $^{\circ}$F
Work Step by Step
a. Because we know that there is a linear relationship between the temperature measured in X and C, we can write an equation in the form y = mx +b. We take this idea and write the relationship as $^{\circ}$C = m($^{\circ}$X) + b. Since we know that 0 $^{\circ}$X is -10 $^{\circ}$C, we can substitute those values into the general equation above. -10 = 0 + b. Therefore, we know that b is -10. Our new equation becomes $^{\circ}$C = m($^{\circ}$X) - 10. We also know that 130 $^{\circ}$C is 50 $^{\circ}$X. Plug these values into the equation again and we get that 130 = m(50) - 10. Add 10 to both sides of the equation to get 140 = 50m. To solve for m, we divide both sides by 50, which gives m = $\frac{14}{5}$. Therefore, the complete relationship between $^{\circ}$C and $^{\circ}$X is $^{\circ}$C = $\frac{14}{5}$($^{\circ}$X) - 10.
b. Using the relationship that we created in part a, we can substitute in 22 $^{\circ}$C into it to result in this equation: 22.0 = $\frac{14}{5}$($^{\circ}$X) - 10. We add 10 to both sides to get this: 32.0 = $\frac{14}{5}$($^{\circ}$X). We then multiply both sides by $\frac{5}{14}$ to solve for $^{\circ}$X, which results in $\frac{80}{7}$ or 11.4.
c. First, convert the $^{\circ}$X to $^{\circ}$C using the exact same process in part b. After substituting 58 $^{\circ}$X into the relationship, we end up calculating that it is equivalent to 152.4 $^{\circ}$C. In order to calculate the temperature in Fahrenheit, recall the equation that relates Celsius to Fahrenheit: $^{\circ}$F = $\frac{9}{5}$($^{\circ}$C) + 32. Plugging in the 152.4 $^{\circ}$C into that relationship, we get that the temperature is also 306.3 $^{\circ}$F. In order to calculate the temperature in Kelvin, we must know the fact that Kelvin is just the temperature in Celsius plus 273.15. Hence, the temperature in Kelvin is 152.4 + 273.15 or 425.6 $^{\circ}$K.
