Math 11 Chapter 1

Solve each problem and show work please

Problems 1 through 4: Regina's Jewelry Regina has a hobby of making jewelry by hand and would like to turn her passion into a business. She plans to sell her work on an ecommeroe website, but she wants to make sure that she is being efcient with her time and does not produce too many or too few pieces. Problem 1 (Related Exercises: Section 1.3 #1, 4, and 5} Each month, Regina is going to make 30 pieces ofjewelry for fun, even if she does not sell any of them. She is willing to make one additional piece each month for every 55 that the selling price increases. This determines the following supply function: 30:) = 30 + 0.2,: Determine the slope and yintercept of this line. In addition, explain how someone would determine this supply function using the Information provided about Reg] na's business. Problem 2 (Related Exercises: Section 1.3 #3") Regina does some research to see how other online jewelry retailers perform. She nds two retailers who make similar jewelry: one retailer sells a quantity of48 pieces per month at a price of $30, and the other sells a quantity ofl pieces per month at a price of 5100. Use this information to write a linear equation that models the quantity demanded as a function of price. 4:001) Problem 3 (Related Exercises: Section 1.3 #9 and 13) Illustrate the supplyfunction from problem 1 and the demand function from problem 2 in a single graph. Your graph should contain the following: I A label indicating what each axis represents II The supply function, labeled S[p] - The demand function, labeled Dip) - An indication of the equilibrium point Problem 4 {Related Exercises: Section 1.3 #13} Based on the information provided, how many pieces should Regina make each month, and at what price should she sell each piece to achieve equilibrium? Explain how you determined your answers. Problems 5 through 7: Cost and Revenue In manufacturing, the cost of production may increase as a business tries to scale to larger volumes of product - the business has more employees to pay, needs to buy more equipment, and may need to buy or lease more facilities to handle the mass production. In some cases, the relationship between cost and quantity can be a quadratic function. Suppose that a small business owner makes a product with the following monthly cost function: (\"(4') = 44' + 242 Here, Crepresents the cost of producing q many units of the product. In addition, market research suggests that the following linear function can be used to predict the monthly quantity demanded based on price p: D(p) = 20 - 0.5p Problem 5 (Related Exercises: Section 1.2 #7 and Textbook Section 1.2 Example 1) Since D(p) represents the demanded quantity of the product, we can let q=D(p). In doing so, we can consider the composite function: C(D(p)) Interpret the meaning of this function. What does it represent in the context of business? Problem 6 (Related Exercises: Section 1.5 #7 and 8) Revenue is the amount of money a business makes from selling their product. If a business makes a quantity q of their product and sells them all at a price of p, the total revenue would be: R =q . P To ensure that all the product inventory is sold, the business should make enough of the product to meet the demand function; thus, the monthly revenue as a function of price becomes: R(p) = D(p) . P For our specific example, the revenue function would be: R(p) = (20 - 0.5p) . p Find the maximum revenue. At what price should the business sell the product to achieve this revenue? Problem 7 (Requires algebra and factoring; graphing might also help) The break-even point is the point at which cost of production and revenue are equal. Mathematically, this can be represented: C(D(p)) = R(p) Here, we are considering both cost and revenue as functions of the selling price. What is the minimum selling price needed break-even? Problem 8: Compound Interest Interest is a fee for borrowing money. The amount of interest charged is typically based on how long money is borrowed (called the term, t), how much money was borrowed (called the principal, P), and the yearly interest rate, /. Note that r is typically given as an APR (annual percentage rate) but should be converted to a decimal for the purposes of calculations. When you invest money in a savings account, you are effectively letting the bank borrow your money to loan to other customers; thus, they bank pays you interest. Some accounts provide compound interest, where interest is computed at regular intervals and added into the account so that more interest is earned as time goes on. Compound interest depends on the number of times, k, that interest iscompounded each year. The total amount in the account (called future value, A) can be determined using the following formula: kt A =P(1+. k Problem 8 (Related Exercises: Section 1.7 #4 and Section 1.8 #9) Suppose your business needs to set aside $100,000 for an emergency fund. That money is invested in an account with a 1.5% APR that is compounded monthly. How long will it take for the account to reach a total balance of $150,000 or more