You want to start a bakery business to earn money. The total amount of money you will invest is $10,000. You will be selling cakes. It will cost you $14 to make each cake.
a)
Write an inequality that represents the cost of making x amount of cakes should not exceed this investment.
* Do not include the dollar symbol in the inequality
You can copy and paste the inequality sign below in your answer.
≥ < > ≤
The inequality that represents the cost of making x amount of cakes should not exceed this investment is:
14x ≤ 10000
b)
Solve this inequality and choose the graph of the solution.
* Make sure to round to the nearest whole number.
Do not include dollar signs in the solution
To solve the inequality 14x ≤ 10000, we divide both sides of the inequality by 14:
x ≤ 10000/14
To round this to the nearest whole number, we get:
x ≤ 714
The graph of the solution would be a closed dot on the number line at 714, and shading to the left, indicating that x is less than or equal to 714.
c)
You plan to sell each cake for $50. Write an equation that represents your total earnings, T, based on the price of your cake and how many you sell. Let x represent the number of cakes you sell.
* Use T and x as your variables
The equation that represents your total earnings, T, based on the price of your cake and how many you sell is:
T = 50x
d)
Using your answer from part b and your equation from part c, what is the most money you can hope to earn from your business?
From part b, we found that the maximum number of cakes you can make is 714 given the investment of $10,000.
Using the equation from part c (T = 50x), we can calculate the maximum amount of money you can hope to earn:
T = 50 * 714
T = $35,700
Therefore, the most money you can hope to earn from your business is $35,700.
e)
Don’t forget that at the beginning of the process you had to spend some of your own money to get started. With the costs taken into account, what was your total profit?
To determine the total profit, we need to subtract the total cost from the total earnings.
The total cost is the cost per cake multiplied by the number of cakes made:
Total cost = 14 * 714
Total cost = $9,996
The total earnings can be calculated using the equation T = 50x, where x is the number of cakes sold.
Total earnings = 50 * 714
Total earnings = $35,700
Total profit = Total earnings - Total cost
Total profit = $35,700 - $9,996
Total profit = $25,704
Therefore, the total profit is $25,704.
Did you make money or lose money
Since the total profit is positive ($25,704), it indicates that you made money in your bakery business.
Your parents heard about the success of your business and they want to help you out. Suppose they want to give you an additional $300 to put toward your business. However, there’s a catch. If you make more than $600 additional profit, you have to give them 10% of your earnings above $600. For example, if you have earned $650, then you owe 10% of $50, or $5, to your parents.
Now that you have an additional $300, revise your inequality from part a to reflect your new spending limit. Solve this inequality.
* Do not include the dollar symbol in the inequality
To revise the inequality from part a to reflect the new spending limit, we add the additional $300 to the total investment. Let's call this new spending limit S. The inequality becomes:
14x ≤ S
S = $10,000 + $300
S = $10,300
The revised inequality is:
14x ≤ $10,300
To solve this inequality, we divide both sides by 14:
x ≤ $10,300/14
Rounding to the nearest whole number, we have:
x ≤ 736
Therefore, the revised inequality is:
x ≤ 736
g)
If you still sell your item for the same price, what is the most money you can hope to earn from your business now?
If you still sell each cake for the same price of $50, the most money you can hope to earn from your business now can be calculated by multiplying the price per cake by the maximum number of cakes you can sell, which is 736:
Total earnings = $50 * 736
Total earnings = $36,800
Therefore, the most money you can hope to earn from your business now is $36,800.
determine how much you will owe your parents.
To determine how much you will owe your parents based on the additional profit, we first need to calculate the additional profit. The additional profit is the total profit minus the original profit, considering the additional investment from your parents.
Original profit = Total earnings - Total cost (from part e) = $25,704
Additional investment = $300
New profit = Total earnings - Total cost - Additional investment = $36,800 - $9,996 - $300 = $26,504
Now, let's calculate the amount of additional profit above $600. If the new profit is more than $600, you owe your parents 10% of the earnings above $600.
Additional profit above $600 = New profit - $600 = $26,504 - $600 = $25,904
Amount owed to parents = 10% of additional profit above $600
Amount owed to parents = 10/100 * $25,904 = $2,590.40
Therefore, you will owe your parents $2,590.40.
i)
You are trying to decide if it is worth the investment to make the cakes if it takes 2.5 hours to make each cake and you have 200 hours this summer to devote to creating your product. Write an inequality that represents your time constraint.
*Use x as the variable
The inequality that represents your time constraint is:
2.5x ≤ 200
j)
Solve your inequality from part i
To solve the inequality 2.5x ≤ 200, we divide both sides of the inequality by 2.5:
x ≤ 200/2.5
x ≤ 80
Therefore, you can make a maximum of 80 cakes within the constraint of having 200 hours this summer.
k)
Remember it costs $14 to make the cakes and you are selling them for $50 each. If you only spend the 200 hours making the cakes, what was your maximum total profit?
To calculate the maximum total profit, we need to consider the number of cakes that can be made within the time constraint and calculate the total earnings and total cost.
From part j, we found that the maximum number of cakes you can make within the 200-hour constraint is 80.
Total earnings = Price per cake * Number of cakes
Total earnings = $50 * 80
Total earnings = $4000
Total cost = Cost per cake * Number of cakes made
Total cost = $14 * 80
Total cost = $1120
Total profit = Total earnings - Total cost
Total profit = $4000 - $1120
Total profit = $2880
Therefore, the maximum total profit you can achieve, by spending the 200 hours making cakes, is $2880.
Does the investment you make and the time constraint allow you to profit from this venture?
Yes, I can make more than $10,000
No, I make less than $10,000
Yes, based on the calculations, the investment and time constraint allow you to profit from this venture. The maximum total profit you can achieve is $2,880, which is more than the initial investment of $10,000.
f)
Now that you have an additional $300, revise your inequality from part a to reflect your new spending limit. Solve this inequality.
* Do not include the dollar symbol in the inequality
You can copy and paste the inequality sign below in your answer.
≥ < > ≤
*
1 point
x≤10,300
To revise the inequality from part a to reflect the new spending limit, we need to include the additional $300 in the investment.
The revised inequality is:
$10,000 + $300 = $10,300
Therefore, the revised inequality is:
x ≤ 10,300
If so, determine how much you will owe your parents.
$2,590.40
Apologies for the confusion in the previous response.
To determine how much you will owe your parents based on the additional profit, we need to calculate the additional profit. The additional profit is the total profit minus the original profit, considering the additional investment from your parents.
Original profit = Total earnings - Total cost (from part e) = $25,704
Additional investment = $300
Total profit with the additional investment = Total earnings - Total cost - Additional investment = $36,800 - $9,996 - $300 = $26,504
Now, let's calculate the amount of additional profit above $600. If the new profit is more than $600, you owe your parents 10% of the earnings above $600.
Additional profit above $600 = Total profit with the additional investment - $600 = $26,504 - $600 = $25,904
Amount owed to parents = 10% of additional profit above $600
Amount owed to parents = 10/100 * $25,904 = $2,590.40
Therefore, you will owe your parents $2,590.40.