Math Snap

STEP 1

Assumptions1. The weekly cost $C$ of producing $x$ units is given by $C(x)=50 x+2250$
. The number of units $x$ produced in $t$ hours is given by $x(t)=60 t$
3. We are looking for $(C \circ x)(t)$, the cost of production as a function of time4. We are also looking for the cost of units produced in4 hours5. Finally, we are looking for the time that must elapse in order for the cost to increase to \(\)15,000$

STEP 2

First, we need to find $(C \circ x)(t)$, which is the composition of the functions $C(x)$ and $x(t)$. This is done by replacing $x$ in $C(x)$ with $x(t)$.
$$(C \circ x)(t) = C(x(t))$$

STEP 3

Now, plug in the given functions for $C(x)$ and $x(t)$.
$$(C \circ x)(t) = C(60t) =50(60t) +2250$$

STEP 4

implify the equation.
$$(C \circ x)(t) =3000t +2250$$This represents the cost of production as a function of time.

STEP 5

Next, we need to find the cost of the units produced in4 hours. This can be done by plugging $t =4$ into $(C \circ x)(t)$.
$$Cost = (C \circ x)(4) =3000(4) +2250$$

STEP 6

Calculate the cost.
Cost =3000(4) +2250 = \($\)12,000 + \($\)2,250 = \($\)14,250So, the cost of the units produced in4 hours is \(\)14,250$.

STEP 7

Finally, we need to find the time that must elapse in order for the cost to increase to \(\)15,000$. This can be done by setting$(C \circ x)(t)$equal to$ $15,000$and solving for$t$.
$$3000t +2250 =15000$$

STEP 8

Subtract2250 from both sides of the equation.
$$3000t =15000 -2250 =12750$$

STEP 9

Divide both sides of the equation by300 to solve for $t$.
$$t =12750 /300$$

Calculate the time.
$$t =12750 /3000 =4.25 \, hours$$So,4.25 hours must elapse in order for the cost to increase to \(\)15,000$.