Question 1

A product’s demand price is \(P(x) = 50 - 2x\) dollars. Revenue is \(R(x) = x \space p(x)\) for \(0 < x < 25\). Which \(x\) maximizes revenue?

A. 12.5

B. 25

C. 0

D. 10

E. 15

Revenue <- function(x) {
  -(x * (50 - 2 * x)) # negative sign for maximization, positive sigh for minimization
}
optimization_result <- optim(par = 10, # parameter: usually middle of interval
                             fn = Revenue, # function: function to maximize/minimize
                             method = "Brent", # method: Brent method for this problem
                             lower = 0, # lower: lower part of the interval
                             upper = 25) # upper: upper part of the interval
cat("The x that maximizes revenue is:",optimization_result$par,"\n")
## The x that maximizes revenue is: 12.5

We find that answer choice A 12.5 is correct.

Question 2

Solve the following system of equations.

\[x + y + z = 1 \\ 2x + 3y + 2z = 3 \\ x + 2y + 5z = 2\]

q2_data <- data.frame(X = c(1,2,1),
                      Y = c(1,3,2),
                      Z = c(1,2,5),
                      Constants = c(1,3,2))
q2_model <- lm(Constants ~ . - 1,data = q2_data) # "." means all other variables except response variable, -1 to exclude intercept
coef(q2_model)
## X Y Z 
## 0 1 0

Question 3

Which one of the following is NOT a factor of 220?

A. 4

B. 10

C. 11

D. 12

# 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() ──
## ✖ 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
q3_data <- data.frame(Choice = LETTERS[1:4],
                      Factor = c(4,10,11,12),
                      Number = rep(220,4))
correct_answer <- q3_data %>%
  mutate(Correct = Number %% Factor == 0) %>%
  filter(Correct == FALSE) %>%
  pull(Choice)
cat("The correct answer is:",correct_answer,"\n")
## The correct answer is: D

Question 4

Plot the corresponding polygon with the given data below.

# install.packages("tidyverse")
library(tidyverse)
q4_data <- data.frame(x = c(5,-3,-3,5),
                      y = c(4,4,-4,-4))
ggplot(q4_data,aes(x = x,y = y)) +
  geom_polygon(fill = "steelblue",col = "black",lwd = 1.5) +
  geom_point(size = 4) +
  coord_equal() +
  theme_gray()