Math / Algebra

Questionthis sinuatione
43. Error Gnatyeis Describe and corrsect the earor in finding the solurion of the equmation $2 x=$ Eix 4a. Staing A subier lis tryingto decide whether or mor to buy a conts s350. The sikier survild hane to ment sibis with either pass shing in o-der to rnake the seasom pass less expentine tham daily passes?
45. Health Clubs Orieheallo club charges a sisosign-up fee and sias per mouth Amofher club charges a s90 sige-up fee and sesper month For vathat mumber of moneftas is the cost of the cinbs eqpal?
45. Geometry The perimeters of the triangles shonumare equall Find the shide lenghas of each triangle
47. Business Asmall juice company spends $\$ 1200$ per dey on business expenses plus $\$ 110$ per boule of juice they make. They charge $\$ 2.50$ for each bortle of juice they produce. Hiow many bortles of juice wust the company sell in one cay im onder tho equal its daily costs?
48. Spreadsheet You set up a spreadsheet to solue $7(x+1)=3(x-1)$ . a. Does your spreadsheet shone the solution of the equation? b. Between withich two values of $x$ is the solution of the equation? How do you loson? C. For what spreadsheet values of $x$ is $7(x+1)$ less than $3(x-1)$ ? \begin{tabular}{|c|c|c|c|} \hline & $A$ & $B$ & $C$ \\ \hline 1 & $x$ & $7(x+1)$ & $3(x-1)$ \\ \hline 2 & -5 & -28 & -13 \\ \hline 3 & -3 & -14 & -12 \\ \hline 4 & -1 & 0 & -6 \\ \hline 5 & 1 & 14 & 0 \\ \hline 6 & 3 & 28 & 6 \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? How many bottles of juice does a small juice company need to sell to cover its daily costs, considering its expenses and selling price? Watch out! Don't forget to include both the fixed business expenses and the per-bottle cost when calculating the total daily cost.

STEP 2

1. Define the cost function
2. Define the revenue function
3. Set up the equation
4. Solve for the number of bottles

STEP 3

Let’s define $C(x)$ as the total daily cost, where $x$ is the number of bottles produced and sold.
The company has fixed daily expenses of $\$1200$ , and it costs them $\$1.10$ to make each bottle.

STEP 4

So, the total cost is the sum of the fixed cost and the variable cost (cost per bottle times the number of bottles): $C(x) = 1200 + 1.10x$

STEP 5

Let’s define $R(x)$ as the total daily revenue.
They sell each bottle for $\$2.50$ , so the revenue is simply the price per bottle multiplied by the number of bottles sold: $R(x) = 2.50x$

STEP 6

To find the break-even point, where the company covers its costs, we need to set the total cost equal to the total revenue: $C(x) = R(x)$

STEP 7

Substituting the expressions we found earlier, we get: $1200 + 1.10x = 2.50x$

STEP 8

Now, let's isolate $x$ to find the number of bottles.
First, subtract $1.10x$ from both sides of the equation: $1200 + 1.10x - 1.10x = 2.50x - 1.10x$ $1200 = 1.40x$

STEP 9

Next, divide both sides by $1.40$ to solve for $x$ : $\frac{1200}{1.40} = \frac{1.40x}{1.40}$ $x \approx 857.14$

STEP 10

Since we can't sell fractions of bottles, we need to round up to the nearest whole number because selling 857 bottles won't quite cover the costs.
So, the company needs to sell 858 bottles to at least cover its daily costs.

STEP 11

The company must sell 858 bottles of juice in one day to equal its daily costs.

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Studdy Solution

STEP 1

What is this asking? How many bottles of juice does a small juice company need to sell to cover its daily costs, considering its expenses and selling price? Watch out! Don't forget to include both the fixed business expenses and the per-bottle cost when calculating the total daily cost.

STEP 2

1. Define the cost function
2. Define the revenue function
3. Set up the equation
4. Solve for the number of bottles

STEP 3

Let’s define $C(x)$ as the total daily cost, where $x$ is the number of bottles produced and sold.
The company has fixed daily expenses of $\$1200$ , and it costs them $\$1.10$ to make each bottle.

STEP 4

So, the total cost is the sum of the fixed cost and the variable cost (cost per bottle times the number of bottles): $C(x) = 1200 + 1.10x$

STEP 5

Let’s define $R(x)$ as the total daily revenue.
They sell each bottle for $\$2.50$ , so the revenue is simply the price per bottle multiplied by the number of bottles sold: $R(x) = 2.50x$

STEP 6

To find the break-even point, where the company covers its costs, we need to set the total cost equal to the total revenue: $C(x) = R(x)$

STEP 7

Substituting the expressions we found earlier, we get: $1200 + 1.10x = 2.50x$

STEP 8

Now, let's isolate $x$ to find the number of bottles.
First, subtract $1.10x$ from both sides of the equation: $1200 + 1.10x - 1.10x = 2.50x - 1.10x$ $1200 = 1.40x$

STEP 9

Next, divide both sides by $1.40$ to solve for $x$ : $\frac{1200}{1.40} = \frac{1.40x}{1.40}$ $x \approx 857.14$

STEP 10

Since we can't sell fractions of bottles, we need to round up to the nearest whole number because selling 857 bottles won't quite cover the costs.
So, the company needs to sell 858 bottles to at least cover its daily costs.

STEP 11

The company must sell 858 bottles of juice in one day to equal its daily costs.

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Studdy Solution

STEP 1

What is this asking? How many bottles of juice does a small juice company need to sell to cover its daily costs, considering its expenses and selling price? Watch out! Don't forget to include *both* the fixed business expenses and the per-bottle cost when calculating the total daily cost.

STEP 2

1. Define the cost function2. Define the revenue function3. Set up the equation4. Solve for the number of bottles

STEP 3

Let’s define C(x)C(x)C(x) as the **total daily cost**, where xxx is the **number of bottles** produced and sold.The company has fixed daily expenses of $1200\$1200$1200, and it costs them $1.10\$1.10$1.10 to make each bottle.

STEP 4

So, the **total cost** is the sum of the fixed cost and the variable cost (cost per bottle times the number of bottles): C(x)=1200+1.10xC(x) = 1200 + 1.10xC(x)=1200+1.10x

STEP 5

Let’s define R(x)R(x)R(x) as the **total daily revenue**.They sell each bottle for $2.50\$2.50$2.50, so the revenue is simply the price per bottle multiplied by the number of bottles sold: R(x)=2.50xR(x) = 2.50xR(x)=2.50x

STEP 6

To find the break-even point, where the company covers its costs, we need to set the **total cost** equal to the **total revenue**: C(x)=R(x)C(x) = R(x)C(x)=R(x)

STEP 7

Substituting the expressions we found earlier, we get: 1200+1.10x=2.50x1200 + 1.10x = 2.50x1200+1.10x=2.50x

STEP 8

Now, let's **isolate** xxx to find the number of bottles.First, subtract 1.10x1.10x1.10x from both sides of the equation: 1200+1.10x−1.10x=2.50x−1.10x1200 + 1.10x - 1.10x = 2.50x - 1.10x1200+1.10x−1.10x=2.50x−1.10x 1200=1.40x1200 = 1.40x1200=1.40x

STEP 9

Next, **divide** both sides by 1.401.401.40 to solve for xxx: 12001.40=1.40x1.40\frac{1200}{1.40} = \frac{1.40x}{1.40}1.401200​=1.401.40x​ x≈857.14x \approx 857.14x≈857.14

STEP 10

Since we can't sell fractions of bottles, we need to round up to the nearest whole number because selling 857 bottles won't *quite* cover the costs.So, the company needs to sell **858 bottles** to at least cover its daily costs.

STEP 11

The company must sell 858 bottles of juice in one day to equal its daily costs.

Get Studdy

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? How many bottles of juice does a small juice company need to sell to cover its daily costs, considering its expenses and selling price? Watch out! Don't forget to include both the fixed business expenses and the per-bottle cost when calculating the total daily cost.

1. Define the cost function
2. Define the revenue function
3. Set up the equation
4. Solve for the number of bottles

Let’s define $C(x)$ as the total daily cost, where $x$ is the number of bottles produced and sold.
The company has fixed daily expenses of $\$1200$ , and it costs them $\$1.10$ to make each bottle.

So, the total cost is the sum of the fixed cost and the variable cost (cost per bottle times the number of bottles): $C(x) = 1200 + 1.10x$

Let’s define $R(x)$ as the total daily revenue.
They sell each bottle for $\$2.50$ , so the revenue is simply the price per bottle multiplied by the number of bottles sold: $R(x) = 2.50x$

To find the break-even point, where the company covers its costs, we need to set the total cost equal to the total revenue: $C(x) = R(x)$

Substituting the expressions we found earlier, we get: $1200 + 1.10x = 2.50x$

Now, let's isolate $x$ to find the number of bottles.
First, subtract $1.10x$ from both sides of the equation: $1200 + 1.10x - 1.10x = 2.50x - 1.10x$ $1200 = 1.40x$

Next, divide both sides by $1.40$ to solve for $x$ : $\frac{1200}{1.40} = \frac{1.40x}{1.40}$ $x \approx 857.14$

Since we can't sell fractions of bottles, we need to round up to the nearest whole number because selling 857 bottles won't quite cover the costs.
So, the company needs to sell 858 bottles to at least cover its daily costs.