Question

SOMEONE PLEASE HELP


quick computing company produces calculators they have found that the cost c(x),of making calculators is a quadratic function in terms of x

the company also discovers that it costs $45 to produce 2 calculators, $143 to produce 4 calculators, and 869 to produce 10 calculators

FIND THE TOTAL COST OF PRODUCING 7 CALCULATORS

Answer 1

You can start with the standard form of a quadratic function, y = ax^2 + bx + c

Then, you can use the three points given to you to create a system of three linear equations

Answer 2

i dont know where to put the numbers in the equation

Answer 3

I will set up the first one for you.

Answer 4

They say it costs $45 to produce 2 calculators. Here 2 is your x and $45 is your y. So:

a(2^2) + b(2) + c = $45 or 4a + 2b + c = $45 is your first equation.

Answer 5

Now, using the second point (x,y) = (4, $143) what equation do you have?

Answer 6

how many equations do i need?

Answer 7

Three, since you have three variables (a, b, and c) that you need to find.

Answer 8

i still dont get where to put them

Answer 9

Your quadratic would look like y = a(x^2) + b(x) + c , and your x and y here are 4 and 143. So, just put in 4 for x and 143 for y ... and what equation do you end up with?

Answer 10

ok thank if i need more help ill ask thank you

Answer 11

Ok :) May I ask a question?

Answer 12

You should end up with three linear equations, and there are about a bazillion ways to solve a system of simultaneous linear equations! I just wondered what method you were wanting to use. For example, have you recently been using Cramer's Rule or working with augmented matrices? It helps to know what approach they want you to take.

Answer 13

I need to go find dinner, but I will check back later just in case you have a question.

Answer 14

@dan815 can someone help?

Answer 15

So, if you use the three points that were given, (2,45); (4,143) and (10,869), you would get these three equations:

45 = a(2^2) + b(2) + c or 45 = 4a + 2b + c; 143 = a(4^2) + b(4) + c or 143 = 16a + 4b + c; and 869 = a(10^2) + b(10) + c or 869 = 100a + 10b + c

Answer 16

Now, we have three linear equations:

45 = 4a + 2b + c
143 = 16a + 4b + c
869 = 100a + 10b + c

What method do you want to use for solving this system?

Answer 17

You can use substitution or elimination. You could also use Cramer's Rule or matrix operations to solve this, but you should probably use whatever method you are currently studying in class.

Answer 18

To use elimination you could start by subtracting the first equation from the second to get an equation involving just a and b:

143 = 16a + 4b + c
-(45 = 4a + 2b + c)
______________________
98 = 12a + 2b

You could further simplify the equation by dividing by 2:

49 = 6a + b

*******************

Now, do the same with a different pair of equations:

869 = 100a + 10b + c
-(45 = 4a + 2b + c )
----------------------
824 = 96a + 8b

You could simplify this equation by dividing by 8, since all of the terms have 8 as a common factor:

103 = 12a + b

Now take your two equations and use elimination again to find a or b.

103 = 12a + b
49 = 6a + b

I would simply subtract the second one from the first one to eliminate b and find a:

103 = 12a + b
-(49 = 6a + b)
_________________
54 = 6a ----> a = 54/6 = 9


Now you have a, go back and find b:

49 = 6a + b --> 49 = 6(9) + b --> 49 = 54 + b ---> -5 = b

-----------------------

So, a = 9 and b = -5. Go back and get one of your original equations to find c:

45 = 4a + 2b + c
45 = 4(9) + 2(-5) + c
45 = 36 -10 + c
45 = 26 + c
19 = c

So now we have a= 9, b= -5, and c = 19

We know it works in the first equation, but need to check to make sure it also works in the other two:

143 = 16a + 4b + c ---> 143 = 16(9) + 4(-5) + 19 --> 143 = 144 -20 + 19 --> 143 = 143 :) Yes!

869 = 100a + 10b + c --> 869 = 100(9) + 10(-5) + 19 --> 869=900-50+19
869=869 :) Yes!

So, then, your quadratic equation is: y = 9x^2 -5x + 19

I hope that is helpful for doing other problems. If there was a different method you wanted to use, please let me know and I'd be happy to give an example using that approach.