Answer
A) $
W= 43.45-0.154t
$
B) $
W(t)=43.45(0.99635)^t
$
The predicted winning time in 2018 is $39.799$ seconds.
Work Step by Step
A)
Let $W$ represent the winning time and $t$ be the number years since 1994
Let $W= b+mt$, then $b = 43.45$. We find the slope from:
$$
m=\frac{\Delta W}{\Delta t}=\frac{40.981-43.45}{16-0}=-0.154 .
$$
The required formula is
$$
W= 43.45-0.154t
$$
B)
Let $W(t)=a(b)^t $ for the exponential function of winning time. The value of $a$ can be taken directly from problem as $a= 43.45$. We now have $W(t)= 43.45(b)^t$. Use the point (16,40.981) to find the value of $b$.
$$
\begin{aligned}
43.45(b)^{16}& =40.981 \\
b^{16} & =\frac{40.981}{43.45}=1.2502\\
b & =(0.943176)^{1 / 16}=0.99635
\end{aligned}
$$
The required formula is
$$
W(t)=43.45(0.99635)^t
$$
The predicted winning time in 2018 is $W=43.45(0.99635)^{24}=39.799$ seconds.
