A home owner goes to the hardware store to purchase the letters that make up a number for a sign he is making, (e.g., F-O-U-R, N-I-N-E, etc.). When he arrives, other customers are doing the same thing. The first customer buys the letters to display ONE, and pays $40. The next customer buys the letters for the number TWO, and pays $60. The last customer buys the letters for the number ELEVEN, and pays $100. Our home owner wants to buy the letters to spell out TWELVE.
The question is: how much does the home owner pay?
ok George, I'll play your game.
I will assume that the store owner is some math-nerd and sets the price of the numbers according to some cubic equation
cost = ax^2 + bx + c, where x is the number of letters and a,b,c are constants
so we have three ordered pairs:
(1,40), (2,60) and (11,100)
for (1,40) ---> a + b + c = 40
for (2,60) ---> 4a + 2b + c = 60
for (11,100) -> 121a + 11b + c = 100
2nd - 3rd: ---> 3a + b = 20
3rd - 1st: ---> 120a + 10b = 60 or 12a + b = 6
subtracting those last two:
9a = -14
a = -14/9
from 12a+b=6
b = 6-12a = 6-12(-14/9) = 74/3
back in 1st: a+b+c=40
-14/9 + 74/3 + c = 40
c = 152/9
so cost = (-14/9)x^2 + (74/3)x + 152/9
so when x = 12, Cost = $88.89
I meant to say:
...according to some quadratic equation
I would need 4 ordered pairs for a cubic.
My work reflects what is needed for a quadratic
Teacher said answer was $120
To determine how much the homeowner pays to spell out TWELVE, we need to analyze the pattern of the previous purchases and calculate the cost per letter.
We already know the prices for the numbers ONE and TWO, which were $40 and $60 respectively.
Let's analyze the difference in cost between the letters in each purchase to determine the individual letter prices.
The difference in cost between ONE and TWO is $60 - $40 = $20, which corresponds to the cost of the letter "T".
Now, let's calculate the cost of each individual letter:
Cost of "O" = Cost of TWO - Cost of "T" = $60 - $20 = $40
Cost of "W" = Cost of ONE + Cost of "O" = $40 + $40 = $80
Cost of "E" = Cost of ELEVEN - 3 * Cost of "O" = $100 - (3 * $40) = $100 - $120 = -$20
Notice that the cost of the letter "E" is negative. This indicates that the homeowner would be receiving a $20 credit for the letter "E". This could be due to an overstock or a promotional deal.
Now, let's calculate the total cost for the letters to spell out TWELVE:
Cost of TWELVE = Cost of "T" + Cost of "W" + Cost of "E" + 2 * Cost of "V"
= $20 + $80 + (-$20) + 2 * Cost of "V"
Since we don't have the cost of the letter "V," we cannot calculate the exact amount the homeowner pays without more information.