Question

The company discovered that it costs $20.50 to produce 3 widgets, $60.50 to produce 7 widgets, and $133 to produce 12 widgets. Using quadratic functions, find the cost of producing 9 widgets.

Answer 1

Do you have access to a graphing calculator... Is it allowed for this problem...

Answer 2

no. all i have is a scientific calculator.

Answer 3

& yes it is allowed. i just don't have one.

Answer 4

Just a minute.

Answer 5

okay

Answer 6

General formula
y = ax^2 + bx + c put in the x,y for each point.
(3,20.50) (7,60.50) (12,133.0)

20.50 = a(3^2) + b(3) + c
60.50 = a(7^2) + b(7) + c
133.00 = a(12^2) + b(12) + c

20.50 = 9a + 3b + c
60.50 = 49a + 7b + c
133.00 = 144a + 12b + c

Subtract equation 1 - equation 2
-40 = -40a - 4b

Subtract equation2 - equation 3
-72.50 = -95a - 5b


-40 = -40a - 4b
-72.50 = -95a - 5b Multiply by 5 and (-4) to get b's the same
5(-40 = -40a - 4b)
-4(-72.50 = -95a - 5b)


-200 = -200a - 20b
290 = 380a + 20b add these

90 = 180a
a = 1/2

plug that into -200 = -200a - 20b
-200 = -200(1/2) - 20b
-200 = -100 - 20b
-100 = -20b
b = 5

plug those into
20.50 = 9a + 3b + c
20.50 = 9(1/2) + 3(5) + c
20.50 = 4.5 + 15 + c
20.50 = 19.5 + c
1 = c

y = 1/2x^2 + 5x + 1

so put in x = 9

y = 1/2(9^2) + 5(9) + 1
y = 1/2*81 + 45 + 1
y = 40.5 + 45 + 1
y = 86.50

Answer 7

If you had a graphing calculator... you put in the three points... find regression and pick quadratic and it will give you the a,b, and c

MUCH QUICKER...

Answer 8

yeah i've realized that. & i don't know anyone who has a graphing calculator which sucks.

Answer 9

but thanks.

Answer 10

If you executed the following expression in Mathematica v8

cost==Fit[{{3,20.50},{7,60.5},{12,133}}, {1,x,x^2}, x] /.x->9

Mathematica 8 running on a mid 2010 IMac with 8Gb ram would spit back with no perceptible delay:

cost==86.5

Right now you could browse over to WolframAlpha.com and paste the following into their "=" box:

Fit[{{3,20.50},{7,60.5},{12,133}}, {1,x,x^2}, x]

Their result would be the cost expression:
\[0.5x^2 + 5.x + 1.\]
When evaluated at x=9 the cost expression would be equal to 86.5 as shown by blexting in his solution.