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 total, 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.
a. Now that you have an additional $300, revise your inequality from part A of
Task 2 to reflect your new spending limit. Solve this inequality and graph the
solution on a number line. Explain what your solution means in terms of the
situation.
b. If you still sell your item for the same price, what is the most money you can
hope to earn from your business now?
c. Will you have to pay your parents? If so, determine how much you will owe
them.
d. Think about how much time it will take you to create your product. You have
200 hours this summer to devote to creating your product. Write an
inequality that represents your time constraint.
e. Solve your inequality from part D and graph your solution on a number line.
Explain what your solution means in terms of the situation.
f. With the costs taken into account, what was your total profit? Did you make
or lose money? Now that you have these values, would you adjust your
business plan from Task 1? If so, how?
a. The revised inequality for the new spending limit is: $x ≤ 600 + 300. Simplifying, this becomes: $x ≤ 900. The solution set for this inequality is all values less than or equal to $900. On a number line, this would be represented as a filled-in circle at 900 and an arrow extending to the left. This means that the maximum amount of money that can be earned without having to give any to the parents is $900.
b. If the item is still sold for the same price, the most money that can be earned from the business now is $900 (the new spending limit).
c. Yes, if the total earnings exceed $600, the parents will have to be paid. To determine how much needs to be paid, we calculate 10% of the amount earned above $600. Let's say the total earnings are $x. If $x > 600, then the amount owed to the parents can be calculated as: 0.10($x - 600).
d. The inequality that represents the time constraint is: 200 ≥ t, where t represents the amount of time (in hours) devoted to creating the product.
e. The solution to the inequality in part d is all values of t that are less than or equal to 200. On a number line, this would be represented as a filled-in circle at 200 and an arrow extending to the left. This means that the maximum amount of time available to create the product is 200 hours.
f. To determine the total profit, we subtract the costs (including the amount owed to the parents) from the total earnings. Let's say the total earnings are $x and the costs are $c (including 10% owed to the parents if applicable). The profit can be calculated as: $x - $c. If the profit is positive, it means money was earned. If the profit is negative, it means money was lost. Adjustments to the business plan from Task 1 would depend on the profit. If the profit is positive, the business plan may not need to be adjusted. If the profit is negative, adjustments may be necessary, such as reducing costs or increasing sales.
a. The revised inequality can be written as:
Total earnings ≤ $600 + $300 - $5(earnings - $600)
Simplifying this inequality, we get:
Total earnings ≤ $900 - $5(earnings - $600)
Total earnings ≤ $900 - $5E + $5(600)
Total earnings ≤ $900 - $5E + $3000
Total earnings ≤ $3900 - $5E
To graph the solution, we plot the values on a number line. Since the total earnings must be less than or equal to $3900 minus $5 times the amount earned beyond $600, the solution can be represented by shading everything to the left of $3900 on the number line.
In terms of the situation, this means that the total earnings from the business must not exceed $3900. If the earnings exceed $600, the parents will receive 10% of the earnings above $600.
b. If you still sell your item for the same price, the maximum amount you can hope to earn from your business is $3900. This is because any earnings above $3900 would require giving 10% of the amount earned above $600 to your parents.
c. Yes, you will have to pay your parents if your earnings exceed $600. To determine how much you will owe them, we can subtract $600 from the total earnings and calculate 10% of that amount.
Amount owed to parents = 10% * (Total earnings - $600)
d. The inequality representing the time constraint is:
Time spent ≤ 200 hours
e. Solving the inequality from part d, we see that time spent must be less than or equal to 200 hours. We represent this on a number line by shading everything to the left of 200.
In terms of the situation, this means that the time spent creating the product cannot exceed 200 hours.
f. To calculate the total profit, we need to subtract the costs from the total earnings. Total profit = Total earnings - Costs.
Since the costs are not given in the question, it is not possible to determine the total profit or whether the business made or lost money.
Based on the information given, it is not possible to determine if adjustments to the business plan are necessary or how they should be made. More information, such as the costs and other factors, would be needed to make informed decisions about adjusting the business plan.
a. To revise the inequality, we need to incorporate the additional $300 and the new spending limit. Let's assume that your total earnings are represented by "x". So the revised inequality would be: x + 300 ≤ 600.
To solve this inequality, we can subtract 300 from both sides of the equation:
x ≤ 300.
The solution to this inequality means that your total earnings (x) must be less than or equal to $300 in order to stay within the new spending limit.
b. Since you still sell your item for the same price and there are no changes in the pricing, the most money you can hope to earn is still $300. This is because any earnings beyond $300 would exceed the new spending limit.
c. No, you will not have to pay your parents in this scenario. According to the revised inequality (x ≤ 300), your total earnings (x) cannot exceed $300, so you will not surpass the $600 threshold that requires you to give them 10% of your earnings above $600.
d. To represent the time constraint of having 200 hours this summer, let's use the variable "t" to represent the time spent creating the product. The inequality would be: t ≤ 200.
This inequality indicates that the total time spent (t) must be less than or equal to 200 hours.
e. To solve this inequality, we can graph the solution on a number line. On the number line, we would mark the point 200 and shade everything to the left of it, representing values that are less than or equal to 200.
This solution means that you cannot spend more than 200 hours creating your product this summer, as it is the upper limit for your time constraint.
f. To calculate the total profit, we need to know the actual earnings from the business. If we assume the total earnings (x) from Task 1, we can calculate the profit by subtracting the costs from the earnings.
Profit = Earnings - Costs
If the total earnings from Task 1 were x, and the cost calculation was given, we could substitute these values into the profit equation to determine the total profit.
If the total profit is positive, it means you made money and your business plan is successful. If the total profit is negative, it means you lost money and you would need to rethink your business plan. Based on the given information, you can determine if adjustments to your business plan are necessary and what changes you should make to improve your financial outcome.