Algebra and Graphing Problem Set

Answer the following questions to the best of your ability. Attempt exercises on your own first to make sure you fully understand the concepts. Feel free to work with anyone in the cohort after giving the problems a try!

Algebra basics

1. Describe how solving for the roots of equations (the values of \(x\) for which a function \(f(x)\) satisfies \(f(x)=0\)) could be useful in environmental science. List at least one method to find roots of a certain kind of funciton.

2. Expand the following expressions:

   1. \((x+2)(3x-4)\)

   2. \((x^2-2y)(x+y^3)\)

3. Solve for \(x\) in the following equations:

   1. \(2x-4=3x+12\)

   2. \(-6ax+2b=x-5c\)

   3. \(2x^2+3x-4=0\)

   4. \(x^2-2x-5=0\)

Story Problems

4. Representative Concentration Pathways (RCP) are climate change scenarios that project future greenhouse gas concentrations in 2100. A new RCP projection indicates that carbon concentrations in the atmosphere will plateau at around 600 ppm. Today’s atmospheric carbon concentration sits at 416 ppm.

   1. If the United States continues to emit proportionally the same amount of carbon (14%), how much carbon will the US emit in 2100 under RCP 4.5? Use the conversion 1 ppm = 2.13 gigaton to give your answer in gigatons.

   2. Let’s say the US decides to pursue an ambitious policy to extract their contribution of carbon from the air. Climatologists model that this policy would have a trajectory of \(f(t)=\frac{-1}{8}t^2+6t+123.83\) where \(t\) is each year and \(f(t)\) is the amount of carbon released by the US. When will the United States eliminate their carbon footprint?

Graphs

6. Find the slope and \(y\)-intercept of the following lines.

x=seq(-5,5)

y=2*x+5

w=(1/3)*x-1

z=x-0.5

u=-3*x+2

p1<-ggplot()+
  geom_hline(yintercept=0)+
  geom_vline(xintercept = 0)+
  geom_point(aes(x=x,y=y),size=3,color="black")+
  geom_line(aes(x=x,y=y),size=2,color="black")+
  theme_bw()+
  theme(axis.line = element_blank())+
  scale_y_continuous(breaks=seq(-5,15,2))+
  scale_x_continuous(breaks=seq(-5,5,by=1))+
  labs(x="",y="")

p2<-ggplot()+
  geom_hline(yintercept=0)+
  geom_vline(xintercept = 0)+
  geom_point(aes(x=x,y=w),size=3,color="blue")+
  geom_line(aes(x=x,y=w),size=2,color="blue")+
  theme_bw()+
  theme(axis.line = element_blank())+
  scale_y_continuous(breaks=seq(-3,3,1))+
  scale_x_continuous(breaks=seq(-5,5,by=1))+
  labs(x="",y="")

p3<-ggplot()+
  geom_hline(yintercept=0)+
  geom_vline(xintercept = 0)+
  geom_point(aes(x=x,y=z),size=3,color="red")+
  geom_line(aes(x=x,y=z),size=2,color="red")+
  theme_bw()+
  theme(axis.line = element_blank())+
  scale_y_continuous(breaks=seq(-5,5,1))+
  scale_x_continuous(breaks=seq(-5,5,by=1))+
  labs(x="",y="")

p4<-ggplot()+
  geom_hline(yintercept=0)+
  geom_vline(xintercept = 0)+
  geom_point(aes(x=x,y=u),size=3,color="darkgreen")+
  geom_line(aes(x=x,y=u),size=2,color="darkgreen")+
  theme_bw()+
  theme(axis.line = element_blank())+
  scale_y_continuous(breaks=seq(-15,15,2))+
  scale_x_continuous(breaks=seq(-5,5,by=1))+
  labs(x="",y="")

cowplot::plot_grid(p1,p2,p3,p4,labels=c("A","B","C","D"))

7. The growth of sharks can be modeled using von Bertalanffy curves. Scalloped hammerhead sharks in the Atlantic were found to have growth curves that follow the graph below.

vb<-function(a,l,k,t0){
  l*(1-exp(-k*(a-t0)))
}

x=seq(0,30)

length=vb(x,214.8,.13,-1.62)

p1<-ggplot()+
  geom_point(aes(x=x,y=length),color="black",size=3)+
  geom_line(aes(x=x,y=length),color="black",size=2)+
  theme_bw()+
  scale_y_continuous(breaks=seq(35,215,5))+
  scale_x_continuous(breaks=seq(0,32,by=2))+
  labs(x="Age (years)",y="Length (cm)")

p1

1. What is the average slope from ages 0 to 8? From 20 to 30?

2. What are the units of the average slopes and why would they change for a shark as they get older?

3. How would you find the instantaneous slope for a shark at age 12? What is the closest approximation you can make with this data and graph?

Picture of a scalloped hammerhead shark.