Question

Widget Wonders produces widgets. They have found that the cost, c(x), of making x widgets is a quadratic function in terms of x.

The company also discovered that it costs $23 to produce 2 widgets, $103 to produce 4 widgets, and $631 to produce 10 widgets. find the cost of producing 6 widgets.

Answer 1

NEED HELP !! D;

Answer 2

this looks painful. they are saying
c(x) = a x^2 + b x + d

and are given (2,23) , (4,103) and (10,631) are points on this parabola.

Answer 3

yea it is painful

Answer 4

do you want to see a picture if thatll help?

Answer 5

to find c(6) we need to find the equation , which means find a, b and d

to do that we use the data they gave us.
using (4,103) we write
103= 16a + 4b + d
using (2,23) we write
23 = 4a + 2b + d

subtract these 2 equations
103= 16a + 4b + d
23 = 4a + 2b + d
------------------
80 = 12a + 2b

simplify (divide all terms by 2)
40 = 6a + b

now do the same thing with
(10,631) and (2,23)

Answer 6

okay

Answer 7

im confused on how you got 16

Answer 8

using (4,103) in
c(x) = a x^2 + b x + d
replace c(x) with 103 and x with 4:
103 = a (4^2) + b*4 + d
4*4 is 16. we get
103 = 16a + 4 b + d (we usually write the letters after the numbers...easier to read)

Answer 9

if they give a graph, perhaps we can read off the answer ?

Answer 10

theres no graph =l

Answer 11

631=20a+10b +d . is that right?

Answer 12

20a? remember is is x^2 which means x*x (not 2*x, which is different)

Answer 13

oh 100a+10b+d

Answer 14

yes
631= 100a+10b+d

now use another equation, example: 23 = 4a + 2b + d (from (2,23) point)

631= 100a+10b+d
23 = 4a + 2b + d

subtract the 2nd equation from the first

Answer 15

608=96a+8b+d ?

Answer 16

d-d is 0

Answer 17

okay just 608=96a+8b

Answer 18

and make our problem easier... divide all terms by 8

Answer 19

76=96+8

Answer 20

i dont know what to do after that

Answer 21

the letters do not disappear. Example a*8/8 = a * 8/8 = a*1= a

96a/8 = 96/8 * a = 12 a

try again

Answer 22

wait where did you get 12

Answer 23

76=96a+8b

Answer 24

you start with
608=96a+8b

you can divide both sides by 8 (and things stay equal)
\[ \frac{608}{8} = \frac{96a+8b}{8} \]
on the right side we can write it as two fractions because
\[ \frac{96a}{8}+\frac{8b}{8} = \frac{96a+8b}{8} \]

\[ \frac{608}{8} = \frac{96a}{8}+\frac{8b}{8} \]
now simplify (and the letters do not disappear)

Answer 25

to make it more clear, it is the same as
\[ \frac{608}{8} = \frac{96}{8}a+\frac{8}{8}b \]

Answer 26

76=12a+1b

Answer 27

yes. we now have two equations with two unknowns. From up above we have
40 = 6a + b
and your equation
76=12a+ b (we can write 1b as just b)
any idea what to do?

Answer 28

uhh. minus it? 36=6a+1b

Answer 29

i have no clue sorry

Answer 30

yes that is the right idea. But do it correctly (carefully)

Answer 31

36=6a+b

Answer 32

I guess you did
76-40 =12a-6a + b - b

?

Answer 33

umm no

Answer 34

omg nvm i read that wrong

Answer 35

when you simplify the b's go away

Answer 36

oh yeah i did

Answer 37

just 36=6a then?

Answer 38

yes. In fact when you see a single b in both equations you should think "aha! subtraction will make the b's go away"

Answer 39

now you have
6a= 36

solve for a (divide both sides by 6)

Answer 40

uhh i just have 6

Answer 41

6a/6 = 36/6
a= 6

Answer 42

but the problem asks the total cost for 6 widgets

Answer 43

yes, but this is a long problem.

however, we are close.
remember what we are doing. they told us we have a quadratic which means we have something that looks like
\[c(x) = a x^2 + b x + d \]
we found a=6
we need to find b and d

but we know
40 = 6a + b
and
76=12a+1b

use either of those equations (with a replaced by 6) to find b
can you do that ?

Answer 44

i dont think so

Answer 45

which equation do you want to use ?

Answer 46

17=12a+1b

Answer 47

i mean 76

Answer 48

ok (because using 17 would not work)
76=12a+1b

we found a=6
erase the a and put in 6 in its place. and remember the equation means
76= 12*a + 1*b

Answer 49

76=72a?

Answer 50

read this carefully:
*erase the a* (no "a"!) and put in 6 in its place and leave everything else as is.
76= 12*a + 1*b

Answer 51

76=72 ..?

Answer 52

you are "doing too much"

erase the "a"
76 = 12 * ___ + 1*b
put 6 in its place
76= 12 * 6 + 1*b

Answer 53

okay

Answer 54

now we can do 12*6 = 72
76 = 72 + 1b

add -72 to both sides

Answer 55

148=144

Answer 56

do it in smaller steps. just write -72 on both sides of the equation
76 = 72 + 1b

Answer 57

sorry i dont have much time

Answer 58

-72 76= 72 + 1b -72

Answer 59

?

Answer 60

yes but we are adding so it should look like this
-72 + 76= 72 + 1b -72
simplify both sides

Answer 61

76-72 on the left side
72-72 on the right side
you get
4 = 0+1b
or just
4 = b

Answer 62

okay

Answer 63

now find d. use
23 = 4a + 2b + d

remember a=6 and b=4

Answer 64

23=24a+8b ?

Answer 65

no, 23 = 4*6 + 2*4 + d

Answer 66

23= 24+8 + d
23= 32 + d

add -32 to both sides
23 - 32 = 32 - 32 + d

Answer 67

55=64+d

Answer 68

your calculator is broken. 23-32 is -9
and
32 - 32 is 0

Answer 69

you should get
d= -9

now you have
c(x) = 6x^2 + 4x - 9

find c(6) which means replace x with 6 and simplify

Answer 70

okay umm

Answer 71

in other words figure out
6* 6^2 + 4*6 - 9

Answer 72

i got 231

Answer 73

231+29-9

Answer 74

is 231 correct?

Answer 75

omg it was! thank u soo much

Answer 76

sorry for probably giving you a headache

Answer 77

yes. it was a tough problem