Math / Algebra

QuestionThe unit cost, in dollars, to produce bins of cat food is $\$ 11$ and the fixed cost is $\$ 3720$ . The revenue function, in dollars, is $R(x)=-2 x^{2}+237 x$ Find the cost function. $C(x)=$
Find the profit function. $P(x)=$
At what quantity is the smallest break-even point? Select an answer

Studdy Solution

STEP 1

What is this asking? We need to find the equations for the cost and profit of making cat food, then figure out how many bins we need to sell to break even! Watch out! Don't mix up cost, revenue, and profit!
Also, remember that "break-even" means profit is zero.

STEP 2

1. Define the cost function
2. Define the profit function
3. Find the break-even point

STEP 3

Alright, let's define the cost function, $C(x)$ !
The cost to make something has two parts: the fixed costs and the variable costs.
Fixed costs are always there, no matter how much you produce.
In this case, the fixed cost is $\$3720$ .
Variable costs change depending on how much you produce.
Here, the unit cost is $\$11$ per bin.

STEP 4

So, if we make $x$ bins of cat food, the variable cost is $11 \cdot x$ , or $11x$ .
To get the total cost, we add the fixed cost and the variable cost:
$C(x) = 11x + 3720$

STEP 5

Now, let's define the profit function, $P(x)$ !
Profit is the money left over after subtracting the costs from the revenue.
We're given the revenue function, $R(x) = -2x^2 + 237x$ , and we just found the cost function, $C(x) = 11x + 3720$ .

STEP 6

So, the profit function is:
$P(x) = R(x) - C(x)$ $P(x) = (-2x^2 + 237x) - (11x + 3720)$ $P(x) = -2x^2 + 237x - 11x - 3720$ $P(x) = -2x^2 + 226x - 3720$

STEP 7

Time to find the break-even point!
This is where the profit is zero, meaning we've made back our costs but haven't made any extra money yet.
So, we need to solve $P(x) = 0$ :
$-2x^2 + 226x - 3720 = 0$

STEP 8

To make this easier, let's divide everything by $-2$ :
$x^2 - 113x + 1860 = 0$

STEP 9

Now, we can solve for $x$ using the quadratic formula!
Remember, for an equation $ax^2 + bx + c = 0$ , the quadratic formula is:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ In our case, $a = 1$ , $b = -113$ , and $c = 1860$ .

STEP 10

Plugging those values into the quadratic formula, we get:
$x = \frac{-(-113) \pm \sqrt{(-113)^2 - 4 \cdot 1 \cdot 1860}}{2 \cdot 1}$ $x = \frac{113 \pm \sqrt{12769 - 7440}}{2}$ $x = \frac{113 \pm \sqrt{5329}}{2}$ $x = \frac{113 \pm 73}{2}$

STEP 11

This gives us two possible solutions:
$x = \frac{113 + 73}{2} = \frac{186}{2} = 93$ $x = \frac{113 - 73}{2} = \frac{40}{2} = 20$ Since the problem asks for the smallest break-even point, our answer is $x = 20$ .

STEP 12

$C(x) = 11x + 3720$ $P(x) = -2x^2 + 226x - 3720$ Smallest break-even point: $20$ bins.

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QuestionThe unit cost, in dollars, to produce bins of cat food is $\$ 11$ and the fixed cost is $\$ 3720$ . The revenue function, in dollars, is $R(x)=-2 x^{2}+237 x$ Find the cost function. $C(x)=$
Find the profit function. $P(x)=$
At what quantity is the smallest break-even point? Select an answer

Studdy Solution

STEP 1

What is this asking? We need to find the equations for the cost and profit of making cat food, then figure out how many bins we need to sell to break even! Watch out! Don't mix up cost, revenue, and profit!
Also, remember that "break-even" means profit is zero.

STEP 2

1. Define the cost function
2. Define the profit function
3. Find the break-even point

STEP 3

STEP 4

So, if we make $x$ bins of cat food, the variable cost is $11 \cdot x$ , or $11x$ .
To get the total cost, we add the fixed cost and the variable cost:
$C(x) = 11x + 3720$

STEP 5

Now, let's define the profit function, $P(x)$ !
Profit is the money left over after subtracting the costs from the revenue.
We're given the revenue function, $R(x) = -2x^2 + 237x$ , and we just found the cost function, $C(x) = 11x + 3720$ .

STEP 6

So, the profit function is:
$P(x) = R(x) - C(x)$ $P(x) = (-2x^2 + 237x) - (11x + 3720)$ $P(x) = -2x^2 + 237x - 11x - 3720$ $P(x) = -2x^2 + 226x - 3720$

STEP 7

Time to find the break-even point!
This is where the profit is zero, meaning we've made back our costs but haven't made any extra money yet.
So, we need to solve $P(x) = 0$ :
$-2x^2 + 226x - 3720 = 0$

STEP 8

To make this easier, let's divide everything by $-2$ :
$x^2 - 113x + 1860 = 0$

STEP 9

Now, we can solve for $x$ using the quadratic formula!
Remember, for an equation $ax^2 + bx + c = 0$ , the quadratic formula is:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ In our case, $a = 1$ , $b = -113$ , and $c = 1860$ .

STEP 10

Plugging those values into the quadratic formula, we get:
$x = \frac{-(-113) \pm \sqrt{(-113)^2 - 4 \cdot 1 \cdot 1860}}{2 \cdot 1}$ $x = \frac{113 \pm \sqrt{12769 - 7440}}{2}$ $x = \frac{113 \pm \sqrt{5329}}{2}$ $x = \frac{113 \pm 73}{2}$

STEP 11

This gives us two possible solutions:
$x = \frac{113 + 73}{2} = \frac{186}{2} = 93$ $x = \frac{113 - 73}{2} = \frac{40}{2} = 20$ Since the problem asks for the smallest break-even point, our answer is $x = 20$ .

STEP 12

$C(x) = 11x + 3720$ $P(x) = -2x^2 + 226x - 3720$ Smallest break-even point: $20$ bins.

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Studdy Solution

STEP 1

What is this asking? We need to find the equations for the cost and profit of making cat food, then figure out how many bins we need to sell to break even! Watch out! Don't mix up cost, revenue, and profit!Also, remember that "break-even" means profit is zero.

STEP 2

1. Define the cost function2. Define the profit function3. Find the break-even point

STEP 3

STEP 4

So, if we make xxx bins of cat food, the **variable cost** is 11⋅x11 \cdot x11⋅x, or 11x11x11x.To get the **total cost**, we add the **fixed cost** and the **variable cost**:C(x)=11x+3720C(x) = 11x + 3720C(x)=11x+3720

STEP 5

STEP 6

STEP 7

Time to find the **break-even point**!This is where the profit is zero, meaning we've made back our costs but haven't made any extra money yet.So, we need to solve P(x)=0P(x) = 0P(x)=0:−2x2+226x−3720=0-2x^2 + 226x - 3720 = 0−2x2+226x−3720=0

STEP 8

To make this easier, let's divide everything by −2-2−2:x2−113x+1860=0x^2 - 113x + 1860 = 0x2−113x+1860=0

STEP 9

STEP 10

STEP 11

This gives us two possible solutions:x=113+732=1862=93x = \frac{113 + 73}{2} = \frac{186}{2} = 93x=2113+73​=2186​=93x=113−732=402=20x = \frac{113 - 73}{2} = \frac{40}{2} = 20x=2113−73​=240​=20Since the problem asks for the *smallest* break-even point, our answer is x=20x = 20x=20.

STEP 12

C(x)=11x+3720C(x) = 11x + 3720C(x)=11x+3720 P(x)=−2x2+226x−3720P(x) = -2x^2 + 226x - 3720P(x)=−2x2+226x−3720 Smallest break-even point: 202020 bins.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to find the equations for the cost and profit of making cat food, then figure out how many bins we need to sell to break even! Watch out! Don't mix up cost, revenue, and profit!
Also, remember that "break-even" means profit is zero.

1. Define the cost function
2. Define the profit function
3. Find the break-even point

So, if we make $x$ bins of cat food, the variable cost is $11 \cdot x$ , or $11x$ .
To get the total cost, we add the fixed cost and the variable cost:
$C(x) = 11x + 3720$

Time to find the break-even point!
This is where the profit is zero, meaning we've made back our costs but haven't made any extra money yet.
So, we need to solve $P(x) = 0$ :
$-2x^2 + 226x - 3720 = 0$

To make this easier, let's divide everything by $-2$ :
$x^2 - 113x + 1860 = 0$

This gives us two possible solutions:
$x = \frac{113 + 73}{2} = \frac{186}{2} = 93$ $x = \frac{113 - 73}{2} = \frac{40}{2} = 20$ Since the problem asks for the smallest break-even point, our answer is $x = 20$ .

$C(x) = 11x + 3720$ $P(x) = -2x^2 + 226x - 3720$ Smallest break-even point: $20$ bins.