The cost of manufacturing $x$ clocks is given by $C(x)=50+27 x-x^{2}$. Also, it is known that in $t$ hours the number of clocks that can be produced is given by $x=2 t$, where $1 \leq t \leq 12$. Express $C$ as a function of $\mathrm{t}$
A. $C(t)=50+27 t-2$
B. $C(t)=50+54 t-4 t^{2}$
C. $C(t)=50+54 t-4 t$
D. $C(t)=50+27 t+t^{2}$
Answer
The cost function $C(t)$ is given by \(\boxed{C(t)=50+54 t-4 t^{2}}\).
Step 1 ：The cost of manufacturing $x$ clocks is given by $C(x)=50+27 x-x^{2}$.
Step 2 ：The number of clocks that can be produced in $t$ hours is given by $x=2 t$, where $1 \leq t \leq 12$.
Step 3 ：We are asked to express $C$ as a function of $t$.
Step 4 ：Substitute $x$ with $2t$ in the cost function $C(x)$ to get $C(t)$.
Step 5 ：So, $C(t) = 50+27(2t)-(2t)^{2}$.
Step 6 ：Simplify to get $C(t)=50+54 t-4 t^{2}$.
Step 7 ：The cost function $C(t)$ is given by \(\boxed{C(t)=50+54 t-4 t^{2}}\).
