Question

in 1870 the average ground temperature in paris was 11.8°c. since then it has risen at a nearly constant rate, reaching 13.5°c in 1969. (a) express the temperature t (in °c) in terms of time t (in years), where t = 0 corresponds to the year 1870 and 0 ≤ t ≤ 99. (give your answer in slope - intercept form.) (b) during what year was the average ground temperature 12.9°c? (round your answer down to the nearest year.)

Explanation:

Step1: Find the slope

The two - point form of a line is used. We have the points $(t_1,T_1)=(0,11.8)$ (corresponding to year 1870) and $(t_2,T_2)=(1969 - 1870,13.5)=(99,13.5)$. The slope $m$ of the line $T=mt + b$ is given by $m=\frac{T_2 - T_1}{t_2 - t_1}=\frac{13.5 - 11.8}{99}=\frac{1.7}{99}\approx0.0172$. The $y$-intercept $b = 11.8$ (when $t = 0$). So the temperature function is $T=0.0172t+11.8$.

Step2: Solve for $t$ when $T = 12.9$

Set $T = 12.9$ in the equation $T=0.0172t + 11.8$. Then $12.9=0.0172t+11.8$. Subtract 11.8 from both sides: $12.9−11.8=0.0172t$, so $1.1 = 0.0172t$. Solve for $t$: $t=\frac{1.1}{0.0172}\approx64$.

Step3: Find the year

Since $t = 0$ corresponds to 1870, the year is $1870 + 64=1934$.

Answer:

(a) $T = 0.0172t+11.8$
(b) 1934