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 $29 to produce 2 widgets, $115 to produce 4 widgets, and $757 to produce 10 widgets.

How much does it cost to make 3 widgets?

Answer 1

Answer:

The total cost to make 3 widgets is $64.

Step-by-step explanation:

As cost is give by quadratic function such as $c(x) = ax^2 + bx + d$

As it costs $29 to produce 2 widgets. So,

$c(2) = a(2)^2 + b(2) + d$

$29= 4a+2b+d ....[A]$

As it costs $115 to produce 4 widgets. So,

$c(4) = a(4)^2 + b(4) + d$

$115= 16a+4b+d ....[B]$

As it costs $757 to produce 10 widgets. So,

$c(10) = a(10)^2 + b(10) + d$

$757= 100a+10b+d ....[C]$

In order to find the values of a, b, c and d, we have to equations [A], [B] and [C]

Subtracting Equation [A] from [B] and [B] from [C]

 12a + 2b = 86  ........[D]

 84a + 6b = 642  ........[E]

Multiplying Equation [D] by 3 and subtracting from [E]

84a + 6b + 3(12a + 2b) = 642 - 86

48a = 384

a = 8

Putting value of a = 8 in equation [D]

12(8) + 2b = 86

96 + 2b = 86

2b = -10

b = -5

Substituting the value of a = 8 and b = -5 in Equation [A].

29= 4a+2b+d

29 = 4(8) + 2(-5) + d

29 = 32 - 10 + d

d = 29 + 10 - 32

d = 7

The required form of equation can be obtained by substituting a = 8, b = -5 and d = 7 in the cost equation. So,

$c(x) = 8x^2 - 5x + 7$ is the required form of equation.

Therefore, the total cost to make 3 widgets will be:

Putting x = 3 in $c(x) = 8x^2 - 5x + 7$

$c(3) = 8(3)^2 - 5(3) + 7$

$c(3) = 72 - 15 + 7$

$c(3) = 64$

Hence, the total cost to make 3 widgets is $64.

Keywords: quadratic equation, cost

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