Basic: $1$ unit costs $\$10.00 $. Increases by $\$50 $ every unit. Total cost for $1000$ units?

Question

The cost for $1$ car is $\$10.00$. Every time you buy $1$, the cost increases by $\$50$. What is the cost for $1000$ units. If you have $10$ million dollars, how many units can you buy.

Thanks
Peter

Answer 1

Hint:

• How much money do you spend if you purchase one unit?

You spend $10000$

• How much if you purchase two units?

You spend $10000+10050$

• Three?

$10000+10050+10100$

• Can you come up with a general pattern for if you want to purchase $n$ units?

The $k^{th}$ unit costs $99950+50k$, so the total cost will be $\sum\limits_{k=1}^n \left(99950+50k\right)$

What do you know about triangle numbers so that you can simplify the above summation?

Answer 2

The cost for 11 car is $\$10.000$ Everytime you buy 11, the cost increases by $\$50$. What is the cost for 1000 units. IF you have 10 million dollars, how many units can you buy.

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The cost of $x$ cars is $10000x+50\left(\frac{x\left(x+1\right)}{2}\right)$. So now our equation becomes:

$10000x+25x\left(x+1\right)\leq10000000$.

Plugging 1000 into the expression, the cost of 1000 cars is $\$35025000$.

And after solving the inequality, we find that we can produce 462 cars if we have $10 million.