Assignment 1 Solution math PDF

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This is a first assignment for business math ii....

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Faculty of Business and Information Technology

Assignment 1 Solution Value: 5%, Due: Feb. 7th @ 11:59 PM This assignment should be completed individually. We take academic integrity seriously. Any student found to be involved in plagiarism or cheating will be penalized in accordance to the Ontario Tech University Calendar, Section 5.15. To assert that you have not given or received, or used unauthorized assistance, write the following pledge in the comment textbox when you submit your files through Canvas system. "I have not given, received, or used any unauthorized assistance." Assignment Instruction: You must submit your detailed workout file to get the full mark on this assignment. Your assignment solution can be typed, scanned, or quality picture. However, if your solution is scanned or picture, YOU MUST PASTE THE SCANNED OR PICTURE DOCUMENTS ONTO ONE MS WORD/PDF FILE. NOTE: Any assignment without valid workout file will receive a grade of ZERO. 1. [10 Points] A plant can manufacture 50 golf clubs per day at a total daily cost of $5159and 60 golf clubs per day for a total cost of $5859. (a) Assuming that daily cost and production are linearly related, find the total daily cost, 𝑪, of producing 𝒙 golf clubs. (b) Graph the total daily cost for 𝟎 ≤ 𝒙 ≤ 𝟐𝟎𝟎. (c) Interpret the slope and 𝒚-intercept of the cost equation. Solution (a) Given: 𝐶1 = 5159,

𝑥1 = 50

𝐶2 = 5859, Slope = 𝑚 =

𝐶2 −𝐶1 𝑥2 −𝑥1

=

𝑥2 = 60 5859−5159 60−50

=

700 10

= 70

Equation of a line: 𝑦 = 𝑚(𝑥 − 𝑥1 ) + 𝑦1 Therefore, 𝐶 = 70(𝑥 − 50) + 5159 = 70𝑥 + 1659 (b) From the equation y-intercept = 1659 and when 𝑥 = 200, C = 15659.

(c) The y-intercept ($1659) is the fixed costs and the slope ($70) is the variable costs per golf club.

2. [12 Points] A truck rental company rents a moving truck for one day by charging $21 plus $0.09 per mile. (a) Write a linear equation that relates the cost 𝑪, in dollars, of renting the truck to the number 𝒙 of miles driven. (b) What is the cost of renting the truck if the truck is driven 129 miles? (c) What is the cost of renting the truck if the truck is driven 483 miles? Solution (a) Base cost = $21 Cost per mile = $0.09 Let 𝑥 be the number of miles travelled per day, then 𝐶(𝑥) = 21 + 0.09𝑥 (b) 𝐶(129) = 21 + (0.09)(129) = $32.61 (c) 𝐶(483) = 21 + (0.09)(483) = $64.47

3. [10 Points] Find the equilibrium point for the pair of demand and supply functions. Here 𝑞 represents the number of units produced, in thousands, and 𝑥 is the price, in dollars. Demand: 𝑞 = 12,000 − 60𝑥

Supply: 𝑞 = 200 + 60𝑥

Solution At equilibrium →

Demand = Supply 12000 − 60𝑥 = 200 + 60𝑥 12000 − 200 = 60𝑥 + 60𝑥 11800 = 120𝑥

11800 = 98.33 120 Now sub 𝑥 = 98.33 into the demand supply equation to compute the number of units produce/sold. 𝑥=

𝑞 = 12000 − 60𝑥 𝑞 = 12000 − (60)(98.33) 𝑞 = 6100.2 ≈ 6100 Therefore, an equilibrium is achieved when 5365 thousand units are produced and sold at a price of $115.88.

4. [15 Points] The table below shows the increase in average monthly price for cable television in the United States since the year 2012. (a) Using Excel, find a linear regression model for the average monthly price, where 𝒙 is the number of years since 2012. (b) Interpret the slope of the model. (c) Use the model to predict the average monthly price in 2023. Year

Average Monthly Price (dollars)

2012 2013 2014 2015 2016 2017 2018 2019

$92.08 $93.58 $96.52 $98.59 $100.23 $101.96 $103.17 $104.76

Solution (a) To find the linear regression equation, enter the data into Excel and follow these steps. i. Select the data range and insert a Scatter chart. ii.

Left click any of the data points on the chart and select Add Trendline from the list.

iii.

From the Trendline Options, select Linear and put a check mark on Display Equation on chart checkbox.

(b) 𝑦 = 1.8413𝑥 + 92.417 Slope = 1.84 implies the average monthly price for cable television in the United States increases $1.84 per year (c) Since year 2023 implies 𝑥 = 11 𝑦 = 1.8413 ∗ 11 + 92.417 = 112.67 Therefore, the average monthly price for cable television in the United States in year 2023 will be $112.67.

5. [15 Points] The table below shows the per capita consumption of ice cream selected years since 1990. (a) Let 𝒙 represent the number of years since 1990 and use Excel to find a cubic regression polynomial for the per capita consumption of ice cream. (b) Use the polynomial model from part (a) to estimate (to the nearest tenth of a pound) the per capita consumption of ice cream in 2025. Year 1990 1995 2000 2005 2010 2015 2020

Ice cream (pounds) 18.1 18.8 15.3 14.7 15.7 14.6 12.9

Solution (a) To find the cubic regression equation, enter the data into Excel and follow these steps. i. Select the data range and insert a Scatter chart. ii.

Left click any of the data points on the chart and select Add Trendline from the list.

iii.

From the Trendline Options, select Polynomial order 3, and put a check mark on Display Equation on chart checkbox.

(b) Since year 2025 implies 𝑥 = 35 𝑦 = −0.0003𝑥3 + 0.0155𝑥 2 − 0.3698𝑥 + 18.681 𝑦 = −0.0003(35)3 + 0.0155(35)2 − 0.3698(35) + 18.681 = 11.863 Therefore, the estimated per capita consumption of ice cream in 2025 is 11.86 pounds.

6. [16 Points] Let 𝐶(𝑥) be the cost to produce 𝑥 widgets, and let 𝑅(𝑥) be the revenue, where 𝒙 is the number of widgets produced and sold. 𝑅(𝑥) = −100𝑥 2 + 1000𝑥,

𝐶(𝑥) = 1000 + 150𝑥

(a) Graph both functions for 0 ≤ 𝑥 ≤ 20. You can use Desmos for your plot. (b) Find the break-even quantities, (c) Find the maximum revenue, and (d) Find the maximum profit. Solution (a) Using Desmos:

(b) At break-even 𝑅(𝑥) = 𝐶(𝑥)

−100𝑥 2 + 1000𝑥 = 1000 + 150𝑥

−100𝑥 2 + 1000𝑥 − 150𝑥 − 1000 = 0 −100𝑥 2 + 850𝑥 − 1000 = 0 𝑎 = −100, 𝑏 = 850, 𝑎𝑛𝑑 𝑐 = −1000 Now, use the quadratic equation formula to solve for 𝑥. −𝑏 ± √𝑏 2 − 4𝑎𝑐 𝑥= 2𝑎 𝑥=

−850 ± √8502 − 4(−100)(−1000) 2(−100) 𝑥1 = 1.41,

𝑥2 = 7.09

(c) Since 𝑅(𝑥) is a parabola opening downwards, the maximum is located at its vertex: 𝑎 = −100, 𝑏 = 1000, 𝑐 = 0 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑐 −

−𝑏 2 10002 = 2500 =0− (4)(−100) 4𝑎

(d) 𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥) 𝑃(𝑥) = −100𝑥 2 + 1000𝑥 − (1000 + 150𝑥) = −100𝑥 2 + 850𝑥 − 1000 𝑎 = −100, 𝑏 = 850, 𝑐 = −1000 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑐 −

−𝑏 2 8502 = 806.25 = −1000 − (4)(−100) 4𝑎

7. [12 Points] A charter flight charges a fare of $300 per person plus $20 per person for each unsold seat on the plane. The plane holds 100 passengers. Let 𝒙 represent the number of unsold seats. (a) Find an expression for the total revenue received for the flight 𝑹(𝒙). Plot the graph. You can use Desmos for your plot. (b) Find the number of unsold seats that will produce the maximum revenue. (c) Find the maximum revenue. Solution (a) Let 𝑥 represent the number of unsold seats, then Revenue = Fare per person * # of seats sold 𝑅(𝑥) = (300 + 20𝑥)(100 − 𝑥) = −20𝑥 2 + 1700𝑥 + 30000

(b) 𝑅(𝑥) = −20𝑥 2 + 1700𝑥 + 30000 𝑎 = −20, 𝑏 = 1700, 𝑐 = 30000 Since 𝑅(𝑥) is a parabola opening downwards, the maximum is located at its vertex: 𝑥=

−𝑏 2𝑎

1700

= − (2)(−20) = 42.5 ≈ 43

Therefore, 43 unsold seats would maximize the revenue of the chartered flight. (c) 𝑅(𝑥) = −20𝑥 2 + 1700𝑥 + 30000 𝑅(43) = −20(43)2 + 1700(43) + 30000 = $66,120 Therefore, the maximum revenue is $66,120

8. [10 Points] A company is planning to manufacture snowboards. The fixed costs are $400 per day and total costs are $5400 per day at a daily output of 20 boards. (a) Assuming that the total cost per day, 𝐶(𝑥), is linearly related to the total output per day, 𝑥, write an equation for the cost function. (b) The average cost per board for an output of 𝑥 boards is given by 𝐶 (𝑥) = 𝐶(𝑥)/𝑥. Find the average cost function. (c) Sketch the graph of the average cost function, including any asymptotes, for 1 ≤ 𝑥 ≤ 30. You can use Desmos for your plot. (d) What does the average cost per board tend to as production increases? Solution (a) Let 𝑥 be the number of snowboards manufactured, then 𝐶(20) = 5400 Fixed costs = 400 Total costs = Fixed costs + Variable costs * quantity manufactured Now, use the information provided for manufacturing 20 snowboards to compute the variable costs (VC). 5400 = 20 𝑉𝐶 + 400 5000 = 20 𝑉𝐶 𝑉𝐶 =

5000 = 250 20

Therefore, 𝐶(𝑥) = 250𝑥 + 400 𝐶(𝑥) 400 250𝑥+400 (b) 𝐶 (𝑥) = = 250 + 𝑥 = 𝑥 𝑥

(c)

400

(d) As the production increases, the 𝑥 term gets smaller and it approaches 0. Therefore, as the production increases the average cost per snowboard will be around $250....