If company A manufactures t-shirts and sells them to retailers for US$9.80 each. It has fixed costs of $2625 related to the production of the t-shirts, and the production cost per unit is US$2.30. Company B also manufactures t-shirts and selll them directly to consumers. The demand for its product is p = 15 −x 125 , its production cost per unit is US$5.00 and its fixed cost are the same as for company A . (i) Derive the total revenue function, R(x) for company A. (ii) Derive the total cost function, C(x) for company A. (iii) Derive the profit function,Π(x) for company A.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
1. If company A manufactures t-shirts and sells them to retailers for US$9.80 each. It has fixed costs of $2625 related to the production of the t-shirts, and the production cost per unit is US$2.30. Company B also manufactures t-shirts and selll them directly to consumers.
The demand for its product is p = 15 −x
125 , its production cost per unit is US$5.00 and its fixed cost are the same as for company A .
(i) Derive the total revenue function, R(x) for company A.
(ii) Derive the total cost function, C(x) for company A.
(iii) Derive the profit function,Π(x) for company A.
(iv) Using a spreadsheet, create a table for showing x, R(x)?, C(x) for company A in the domain x = 50, 100, 150, 200, 250, 300, 350, 400, 450.
(v) Graph the functions from (d) above on the same axes.
(vi) From your graph, determine the break-even level of output for company A.
(vii) Derive the total revenue function,R(x) for company B.
(viii) Derive the profit function,Π(x) for company B.
(ix) How many t-shirts must company B sell to in order to break-even.
(x) How many t-shirts must company B sell to maximise its profit.
? (a) A company has determined that its profit for a product can be described by a linear function. The profit from the production and sale of 150 units is $455, and the profit from 250 units is $895.
(i) What is the average rate of change of the profit for this product when
between 150 and 250 units are sold?
(ii) Write the equation of the profit function for this product.
(iii) How many units give break-even for this product?
(b) You are the CEO for a lightweight compasses manufacturer. The demand function for the lightweight compasses is given by
p = 40 − 4q2 where q is the number of lightweight compasses produced in millions.It costs the company $15 to make a lightweight compass.
(i) Write an equation giving profit as a function of the number of lightweight compasses produced.
(ii) At the moment the company produces 2 million lightweight compasses and makes a profit of $18,000,000, but you would like to reduce production. What smaller number of lightweight compasses could the company produce to yield the same profit?
? (a) If Daniella opens a mutual fund account with a deposit of $15,000 and it promises to pay an annual interest rate of r%, compounded on a monthly basis.
(i) Derive a function B, which shows the balance on Daniella′s account the end of 6 years.
(ii) Make a comparison of the rates of change in B with resect to the interest rate when r is 8% , 10% and 12%.
(b) A virus is believed to be spreading through a sheep farm according to that equationN = 450 (1
1 + 249e−0.1t ) where N is the number of sheep infected and t is the number of days.
(i) How many sheep are infected at t = 0?
(ii) Using a spreadsheet calculate the number of sheep infected after 0, 10, 30 , 50, 70, 100 days)
(iii) Will all the sheep become infected eventually? Give reasons to support your answer.
? (a) The service elevator at AR Hector Financial Complex has a maximum capacity of 2870 lbs. If the elevator operator weighs 205 lbs and has to move hundreds of boxes from the ground floor to the Penthouse, if each box weighs 65 lbs, what is the maximum number of boxes that can be
safely conveyed on the elevator on each trip?
(b) The borough of Pan Sherzando charges homeowners $75.24 per month plus $3.86 per 1000 litres for water usage in excess of 10,000 litres. In 2019 one homeowner’s monthly bill ranged from $137.00 to $245.08. Find the range for the water used by that homeowner.
(c) A contractor has to build rectangular pool 12 metres long and 8 metres wide. The pool must, by agreement, contain a rectangular border of uniform width and must be exactly 120 sq meters in area.
(i) How wide should the border be? Round to the nearest tenth of a metre.
(ii) If physical planning laws require at least a 2-metre-wide border around a pool, would the contractor be able to satify both the client and the planning regulators?
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