Your parents heard about the success of your business and they want to help you

Question

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?

Answer 1

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.