Question 1

Convert \(\frac{26 \pi}{5}\) radians into degrees.

# install.packages("pracma")
library(pracma)
answer <- rad2deg(rad = 26 * pi / 5)
cat("26 pi / 5 radians =",answer,"degrees","\n")
## 26 pi / 5 radians = 936 degrees

Question 2

What is the area between the curve \(f(x) = 4x^3\) and the \(x\)-axis from \(x = 0\) to \(x = 1\)?

# install.packages("tidyverse")
library(tidyverse)
## Warning: package 'lubridate' was built under R version 4.5.2
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.5
## ✔ forcats   1.0.1     ✔ stringr   1.5.2
## ✔ ggplot2   4.0.0     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ purrr::cross()  masks pracma::cross()
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
f <- function(x) {
  4 * x^3
}
area <- integrate(f,lower = 0,upper = 1)$value
x_values <- seq(0,1.05,length.out = 500)
y_values <- f(x_values)
q2_data <- data.frame(x = x_values,y = y_values)
ggplot(q2_data,aes(x = x,y = y)) +
  geom_line(col = "black",lwd = 1.25) +
  geom_ribbon(data = subset(q2_data,x >= 0 & x <= 1),
              aes(ymin = 0,ymax = y),
              fill = "red") +
  labs(title = "Graph of f(x) = 4x^3",
       caption = paste("The area under the curve between x = 0 and x = 1 is:",area),
       x = "x",
       y = "y") +
  theme_gray(base_size = 14)

Question 3

Ben takes an ordinary deck of 52 playing cards and removes all the picture cards - all the Jacks, Queens, and Kings. Ben then shuffles the remaining cards and selects a card at random. Ben wins if the number on the card is a prime number. Use a Monte Carlo simulation to estimate the probability of winning.

# install.packages("pracma")
library(pracma) # for the isprime() function
Cards <- 1:10 # 1 represents the Ace, 2 - 10 are normal
counter <- 0 # number of prime number cards
N <- 100000 # number of trials
for (x in 1:N) {
  pick <- sample(x = Cards,size = 1,replace = T)
  if (isprime(pick) == TRUE) {
    counter <- counter + 1
  }
}
probability <- counter / N
cat("The probability of winning is:",probability,"\n")
## The probability of winning is: 0.40209

Question 4

You are given some data about products and prices. Answer the following parts.

Part 1: Define the data frame.

q4_data <- data.frame(Product = c("Salt","Bread","Flour","Toothbrush","Soap","Lollipop","Cake","Tissues"),
                      Price = c(0.50,0.93,0.90,0.75,0.30,0.25,0.48,0.79))
q4_data
##      Product Price
## 1       Salt  0.50
## 2      Bread  0.93
## 3      Flour  0.90
## 4 Toothbrush  0.75
## 5       Soap  0.30
## 6   Lollipop  0.25
## 7       Cake  0.48
## 8    Tissues  0.79

A. Arrange the data in order from least price to greatest price.

# install.packages("tidyverse")
library(tidyverse)
q4_data %>%
  arrange(Price)
##      Product Price
## 1   Lollipop  0.25
## 2       Soap  0.30
## 3       Cake  0.48
## 4       Salt  0.50
## 5 Toothbrush  0.75
## 6    Tissues  0.79
## 7      Flour  0.90
## 8      Bread  0.93

B. Arrange these in order from greatest price to least price.

# install.packages("tidyverse")
library(tidyverse)
q4_data %>%
  arrange(desc(Price))
##      Product Price
## 1      Bread  0.93
## 2      Flour  0.90
## 3    Tissues  0.79
## 4 Toothbrush  0.75
## 5       Salt  0.50
## 6       Cake  0.48
## 7       Soap  0.30
## 8   Lollipop  0.25

C. Arrange this data in alphabetical order.

# install.packages("tidyverse")
library(tidyverse)
q4_data %>%
  slice(str_order(Product))
##      Product Price
## 1      Bread  0.93
## 2       Cake  0.48
## 3      Flour  0.90
## 4   Lollipop  0.25
## 5       Salt  0.50
## 6       Soap  0.30
## 7    Tissues  0.79
## 8 Toothbrush  0.75