Tom Is Going To Paint His Kitchen Ceiling: Math Assignment, TCD, Ireland

Question 1

d) Tom is going to paint his kitchen ceiling. The ceiling is rectangular; its length is 4 meters, and its width is 6 meters. Tom knows 1 liter of paint covers a surface area of 6 square meters.

(i) Calculate the area of Tom’s ceiling.

(ii) How many liters of paint will Tom need to paint 2 coats of paint on his kitchen ceiling?

Question 2 – Statistics

a) State whether the following are examples of continuous data or discrete data:

(i) The time is taken by each student in a class to run 100 meters

(ii) The number of iPhones sold in 2020 2.

b) The ages of the 12 members of a choir are listed below:

From the data above calculate the following:

(i) the median age of the choir members,

(ii) the interquartile range of the choir members.

c) Given the frequency table below of patients in a hospital ward, and using mid-interval values, calculate the following correct to one decimal place:

(i) the mean age of the patients.

(ii) the standard deviation of the distribution.

(iii) Draw a histogram representing the frequency distribution.

(iv) Comment briefly on the skewness of the data.

Question 3 – Probability

a) Two dice are thrown. What is the probability that the total of the 2 dice will be at least 10?

b) A box contains 24 green, 16 blue, and 10 red raffle tickets.

(i) What is the probability a green raffle ticket is chosen?

(ii) What is the probability that a non-red raffle ticket is chosen? (iii) What is the probability a blue or a red raffle ticket is chosen?

c) The combination to Mike’s hotel safe has four digits, and each digit can be any number from 0 to 9. Mike can set the combination himself, but he doesn’t like the number 8 so he refuses to use that number in his combination. How many different combinations to his safe could Mike set

(i) if digits can be repeated,

(ii) if repetition of digits is not allowed?

(iii) if repetition is allowed, and Mike wants his combination to be an even number?

d) Mary rolls a die over and over. She keeps count of the number of sixes she rolls, and they are given in the table below.

(iii) What is the actual probability of rolling a six using a fair die?

(iv) Does the die appear from the graph to be biased towards sixes or biased against sixes? Explain your answer making reference to relative frequency and the actual probability of rolling a six using a fair die.

Question 4 – Functions & Graphs

a) (i) Define the inverse of a function.

(ii) What is the inverse of the function 𝑓(𝓍) = 4 – 5𝓍?

b) L is a line with equation 2𝓍 + 𝑦 = 4.

(i) Find the slope of line L.

(ii) line K is parallel to line L. Line K passes through the point (1, -1). Find the equation of the line K.

(iii) line M is perpendicular to line L. Line M passes through the point (2, 1). Find the equation of the line M. (iv) On the 𝓍-𝑦 plane, graph the line L. On your graph, clearly mark and identify the coordinates of 2 points on the line L.

c) (i) Draw the graph of the function 𝑓: 𝓍 → 2𝓍2 – 3𝓍 – 6 in the domain -2 ≤ 𝓍 ≤ 3.

(ii) Is the function 𝑓 known as a linear, quadratic or square function?

(iii) Find from the graph drawn in (i) the approximate range of values of 𝓍 for which 𝑓(𝓍) ≤ 0. Approximate your answer(s) to 1 decimal place.

Question 5 – Linear Programming  A farmer has a very large farm on which to grow his two crops of potatoes and carrots. The costs he incurs (buying seeds, fertilizer, etc.) when growing potatoes are €100 per acre, and are €200 per acre when growing carrots. He has a total budget of €10,000. He has only 1200 man-hours available to him for planting all his crops. It takes 10 man-hours to plant an acre of potatoes, and it takes 30 man-hours to plant an acre of carrots. At the market, the farmer is always guaranteed to sell out all his crops. He expects that his potatoes will yield a profit of €50 per acre, and his carrots a profit of €120 per acre. Given that the farmer’s objective is to maximize his profit, answer each of the questions below.

(a) Letting 𝓍 = the total area of potatoes (in acres) he should grow, and 𝑦 = the total area of carrots (in acres), write down the objective function for this problem.

(b) Write down the constraints for this problem as a system of 4 inequalities (including the two non-negative constraints).

(c) Graph the constraint inequalities, and on the graph shade in the ‘feasible region’ for this problem.

(d) Write down clearly the coordinates of the 4 points [including the point (0, 0)] that are at the 4 corners (i.e. the vertices) of the feasible region for this problem?

(e) How many acres of potatoes and how many acres of carrots should the farmer grow in order to maximize his profit?

(f) What is the maximum profit that the farmer can make?