Math  /  Algebra

Question47. Business A small juice company spends $1200\$ 1200 per day on business expenses plus $1.10\$ 1.10 per bottle of juice they make. They charge $2.50\$ 2.50 for each bottle of juice they produce. How many bottles of juice must the company sell in one day in order to equal its daily costs?
48. Spreadsheet You set up a spreadsheet to solve 7(x+1)=3(x1)7(x+1)=3(x-1). a. Does your spreadsheet show the solution of the equation? b. Between which two values of xx is the solution of the equation? How do you know? c. For what spreadsheet values of xx is 7(x+1)7(x+1) less than 3(x1)3(x-1) ? \begin{tabular}{|c|c|c|} \hline & AA & BB \\ \hline 1 & xx & 7(x+7(x+ \\ \hline 2 & -5 & -28 \\ \hline 3 & -3 & -14 \\ \hline 4 & -1 & \\ \hline 5 & 1 & \\ \hline 6 & 3 & \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? How many bottles of juice need to be sold for the juice company's income to equal its daily expenses?
Also, does the spreadsheet correctly solve the given equation, between what two *x*-values does the solution lie, and when is 7(x+1)7(x+1) less than 3(x1)3(x-1)? Watch out! Don't mix up cost and income!
Also, remember to check your spreadsheet calculations carefully.

STEP 2

1. Calculate break-even point
2. Complete the spreadsheet
3. Analyze the spreadsheet

STEP 3

Let *C* be the **total daily cost**.
The company spends $1200\$1200 per day plus $1.10\$1.10 per bottle.
If they make *x* bottles, the cost is C=1200+1.10xC = 1200 + 1.10x.

STEP 4

Let *I* be the **total daily income**.
The company charges $2.50\$2.50 per bottle.
If they sell *x* bottles, the income is I=2.50xI = 2.50x.

STEP 5

We want to find the number of bottles *x* where the income *I* equals the cost *C*.
So, we set I=CI = C, which gives us 2.50x=1200+1.10x2.50x = 1200 + 1.10x.

STEP 6

To solve for *x*, first subtract 1.10x1.10x from both sides of the equation: 2.50x1.10x=1200+1.10x1.10x2.50x - 1.10x = 1200 + 1.10x - 1.10x.
This simplifies to 1.40x=12001.40x = 1200.
Now, divide both sides by **1.40**: 1.40x1.40=12001.40\frac{1.40x}{1.40} = \frac{1200}{1.40}.
This gives us x=12001.40857.14x = \frac{1200}{1.40} \approx 857.14.
Since we can only sell whole bottles, we round up to the nearest whole number, which is **858** bottles.

STEP 7

We need to fill in the blanks in the spreadsheet.
For x=1x = -1, 7(x+1)=7(1+1)=7(0)=07(x+1) = 7(-1+1) = 7(0) = 0.
For x=1x = 1, 7(x+1)=7(1+1)=7(2)=147(x+1) = 7(1+1) = 7(2) = 14.
For x=3x = 3, 7(x+1)=7(3+1)=7(4)=287(x+1) = 7(3+1) = 7(4) = 28.

STEP 8

The spreadsheet is meant to help solve 7(x+1)=3(x1)7(x+1) = 3(x-1).
Let's simplify the equation: 7x+7=3x37x + 7 = 3x - 3.
Subtracting 3x3x from both sides gives 4x+7=34x + 7 = -3.
Subtracting 77 from both sides gives 4x=104x = -10.
Dividing both sides by 44 gives x=104=52=2.5x = -\frac{10}{4} = -\frac{5}{2} = -2.5.
The spreadsheet *doesn't* show this solution directly.

STEP 9

The solution x=2.5x = -2.5 is between 3-3 and 1-1, which are in the spreadsheet.
We see that at x=3x = -3, 7(x+1)=147(x+1) = -14 and at x=1x = -1, 7(x+1)=07(x+1) = 0.
Since 3(x1)3(x-1) changes continuously with *x*, and 7(x+1)7(x+1) also changes continuously with *x*, the solution must be between 3-3 and 1-1.

STEP 10

From our simplified equation 4x+7=34x + 7 = -3, we found x=2.5x = -2.5.
When x<2.5x < -2.5, 7(x+1)7(x+1) is less than 3(x1)3(x-1).
Looking at the spreadsheet, this occurs when x=5x = -5 and x=3x = -3.

STEP 11

The juice company must sell **858** bottles to equal its daily costs.
The spreadsheet *doesn't* directly show the solution x=2.5x = -2.5 to the equation 7(x+1)=3(x1)7(x+1) = 3(x-1).
The solution lies between x=3x = -3 and x=1x = -1. 7(x+1)7(x+1) is less than 3(x1)3(x-1) when x<2.5x < -2.5, which in the spreadsheet is when x=5x = -5 and x=3x = -3.

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord