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Mid-Unit Assignment: ​Dragon’s Den

Virtual High School/Kells Academy

Mr. Ken Stewart

Advanced Functions MHF4U

01/16/2020

Question 2.

The product that I am presenting to dragons is a new type of dry shampoo with new ingredients that will make your hair look shinier and cleaner than the dry shampoos currently being produced. It is also an all organic shampoo, free of any harmful chemicals that will appeal to a larger group of consumers compared to the products currently being sold.

Question 3.

The dry shampoos are being produced currently at the cost of $4.5 in my laboratory. In other words, I am producing these products from home and I avoid paying more money on the rent. The function that represents the cost of producing dry shampoos is C(x)=mx which is C(x)=4.5x

Question 4.

Revenue is the amount of money that is made from the products sold without taking expenses into consideration. R(x) is the function used to calculate the amount of revenue for 500 Dry shampoos with a price of $9 per unit per month.

R(x)= (price)*(number of products sold) R(x)= 9x

a. If we consider that for every $0.15 increase in price for Dry Shampoos, 10 fewer units will be sold, we have to write R(x) with respect to that. The number of Dry shampoos sold is represented by (500-10x) and the price of the Dry shampoo is represented by (9+0.15x). Therefore R(x) is shown by: R(x)=(500-10x)(9+0.15x)

b.

c. According to the calculations shown below, the x-intercepts are x=-60 and x=50. The values obtained represent by how much the price has to increase so that the revenue will be zero.

According to the last sentence, we need a valid (positive) number for price increase, which means x=-60 cannot be the price increase because the increase has to be a positive value. Therefore, x=50 is the only valid number that is left. This means that if the increase in the price of the product is 50 cents the profit that I make will be zero. Any value of x (increase in the price of my product) that is lower than 50, would mean that I make some profit.

d. Maximum: the vertex of the function occurred at (-5,4537.5). Therefore, the maximum value that y can have is 4537.5. Which means I will have the maximum amount of revenue if the product is sold 5 cents cheaper than the current price. The largest amount of revenue for a cost of $8.95 per month is $4537.5.

Minimum: according to the graph in part b. The function is downward opening on both 3rd and 4th quadrant. Therefore, it has no minimum point. e. The domain has to be only the positive x(es) if only the increase in price is considered.

The Range is the revenue and it cannot be negative. The largest amount of revenue is $4537.5 in this function.

f.

The Dry Shampoos price will increase over a period of time. If we increase the price from $9 to $9.50, the change in revenue can be modeled using the following function: R(x)=9.50x.

According to the previous calculation, if the price increases from $9 (R1) to $9.50 (R2), the revenue gained for selling 500 Dry Shampoos increases from $4500 to $4750.

Question 5.

In order to find the profit, we have to consider that profit is the money that remains after paying all expenses and costs. Therefore, the function by which profit can be modeled is P(x)=R(x)-C(x). When R(x)=9.50x and C(x)=4.5x.

Based on the calculations above, the profit for Dry Shampoos is 5x and if 500 Dry Shampoos are sold in a month, the profit is $2500.

The graph of P(x)=5x represent the gained profit with respect to the units of Dry Shampoos sold.

Question 6.

Hello Dragons. I am Sayna Javanmardi representing my new Dry shampoo. Have you ever gotten up in the morning before going to work thinking “My hair is a mess! I can’t go to work looking like this!”.

When you have no time to take a shower or dry your hair, this product will come to your rescue. I am here with this fantastic dry shampoo that can save you in all kinds of social and professional situations. This is a small spray that you can just put it in your bag and use whenever you need your hair to look perfect.

It saves time and is cheap, so everyone can afford it. It is a unique product in that it is organic without any harmful chemicals and will make your hair shinier compared to other dry shampoo products.

Currently, I am producing this dry shampoo in my small laboratory at home. I have already patented this shampoo and gotten the FDA approval. I am asking for 100K (100 thousand dollars) for 2% of my company.

I produce this Dry Shampoo at $4.5 per 200mL bottles and sell each at the price of $9.5 which gives me a $5 profit per product.

Question 7.

I believe that my results were realistic enough in this assignment. Based on the calculations, when the price increases by $0.15 it will lead to an increase of $0.50 in the revenue. This is mainly because when the price increases by $0.15 a smaller number of products are sold. This is also realistic because whenever the price of a product increases, the number of customers are likely to decrease.

What I learned from this mid-unit is the relation of formulas and lessons we learn with their real-world application. For example, when buying a product, the calculations behind the business may not come to mind but this assignment made me think about all the work that goes into starting a business. This assignment showed me how minor changes in the price of the product might affect the company and its profit. Although the calculations were simplified, choosing and presenting the product to the Dragon’s Den in a pitch was quite challenging for me.