Math Answers

Prove for all integers a and b, if a mod 8 = 5 and b mod 8 = 3 then ab mod 8 = 7.

Prove for every integer n, n4 has the form 8m or 8m + 1 for some integer m. Hint: Use 2 cases.

In a large population 63% of the people have been vaccinated if 5 people are randomly selected what is the probability that at least one of them has been vaccinated

An interest rate of 14.90% per year compounded every three months is equivalent to a weekly compounded compounded interest rate of

BAT Industries is a manufacturer of orange juice in Jamaica. In 2022 it sold a bottle of orange juice to wholesalers for $100 and its variable costs is $40 and fixed costs is $100,000 monthly:

Using the information provided. 

i. How many bottles of orange juice must the company sell in order to break even and what is the breakeven revenue? (4 marks)

ii. If the company sold 1,500,000 bottles of orange juice for the year 2022, what was its profit? (4 marks)

iii. Due to inflation, in 2023 fixed costs are expected to rise to $175,000 monthly. What will be the NEW break-even quantity and revenue? (4 marks)

iv. If the company will sell the same number of orange juice as it did in 2022 and it wants to make the same profit as in 20222, what price should they sell each bottle of orange Juice for in 2023? (5 marks)

Let X be a binomial(5, 0.2) random variable. Let Y be a discrete random variable that is independent of X, such that Y = 1 with probability 0.2 and Y = 0 with probability 0.8. What is the probability that the sum of X and Y is less than or equal to 3?

Find a general solution of the following differential equation

dy/dx - 5y = (xe^-2x)*y^-2

Solve the given initial value problem.

dy/dx = y(2x^2 +y^2)/2x^3

subject to y(1) = 1.

In a study of life expectancy of 400 people in a certain geographic region, the mean age death was 70 years, and the standard deviation was 5.1 years. If a sample of 50 people from this region is selected, what is the probability that he mean life expectancy will be less than 68 years?