Find the Expression

Question

You need to find a formula for the number of cars produced by a new plant that was opened 25 days ago. You are told that three days ago the plant had 450 cars on hand, and that yesterday it had 480 cars on hand. You also are told that some of these cars are from previous inventory. Assume that the plant produces cars at this same rate and has produced them at this same rate since it opened.

Answer 1

three days ago the plant had 450 cars

yesterday the plant had 480  cars

in two days the plant built 30 cars, or 15 cars per day

the plant has been open for 25 days

three days ago the plant had been open for 22 days

22*15=330 cars built in twenty-two days

450-330=120 cars from previous inventory

120+15d=I, where d=number of days and I=total inventory

 

 

Answer 2

Hi Chantilly.  Tho start this problem, we should look for the most basic equation to describe the plant first.  We are looking for an equation for the number of cars produced, knowing that they plant started with an initial inventory and that new cars are produced at a steady rate.  From this, we can write the very generic equation:

 

C_T = C_I + rd  _{Equation 1}

where C_T= total number of cars, C_I= the initial inventory, r= rate of production/day and d= number of days that have past.

 

We know from the above that C_I and r will have numerical values, which we must solve for and d will be our variable.  We must then find the values for C_I and r.

 

Let's start with the value for r.  We are told that three days ago, we have 450 cars and yesterday (two days later) we have 480 cars.  The difference in these numbers will give us how many cars were produced in two days.  480-450= 30 new cars in two days.  Since r is the rate of production/day, and we know that 30 cars were produced in two days, we can find the rate/day by dividing 30 new cars by 2 days to get 15 cars produced/day.  r = 15 cars/day.

 

Now we can rewrite the original equation with this new rate plugged in for r :

 

C_T = C_I + 15d  _{Equation 2}

 

Next, we must solve for C_I.  We are told two pieces of information that will help us solve for this variable.  First, the plant was opened 25 days ago. Second, that yesterday (24 days after the plant opened), there were 480 cars in inventory.  We can plug these numbers in the equation above and then solve for C_I

 

480 = C_I + (15x24)

 

After you solve for C_I, you can plug that number into Equation 2 and you have your final equation.